LINEAR TOPOLOGICAL SPACES OF CONTINUOUS VECTOR

LINEAR TOPOLOGICAL SPACES OF CONTINUOUS VECTOR-VALUED FUNCTIONS Liaqat Ali Khan

AP Academic Publications

Linear Topological Spaces of Continuous Vector-Valued Functions Authors: Liaqat Ali Khan Address: Department of Mathematics Faculty of Science King Abdulaziz University, Jeddah

DOI: 10.12732/acadpubl.201301 Pages: 350

Typesetting system: LATEX c

Academic Publications, Ltd., 2013 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Academic Publications, Ltd., http://www.acadpubl.eu), except for scientific or scholarly analysis.

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Academic Publications, Ltd., 2013 http://www.acadpubl.eu

Contents Introduction

vi

Preface

vii

Acknowledgements

xiii

Chapter 1. The Strict and Weighted Topologies 1. Strict and related Topologies on Cb (X, E) 2. Weighted Topology on CVb (X, E) 3. Notes and Comments

1 2 20 24

Chapter 2. Completeness in Function Spaces 1. Completeness in the uniform topology 2. Completeness in the Strict Topology 3. Completion of (Cb (X, E), β) 4. Completeness in the Weighted Topology 5. Notes and Comments

27 28 33 38 41 43

Chapter 3. The Arzela-Ascoli type Theorems 1. Arzela-Ascoli Theorem for k and β Topologies 2. Arzela-Ascoli Theorem for Weighted Topology 3. Notes and Comments

45 46 53 61

Chapter 4. The Stone-Weierstrass type Theorems 1. Approximation in the Strict Topology 2. Approximation in the Weighted Topology 3. Weierstrass Polynomial Approximation 4. Maximal Submodules in (Cb (X, E), β) 5. The Approximation Property 6. Notes and Comments

63 64 72 77 79 82 87

Chapter 5. Maximal Ideal Spaces 1. Maximal Ideals in (Cb (X, A), β) 2. Maximal Ideal Space of (Cb (X, A), β) 3. Notes and Comments

89 90 94 99

iii

iv

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Chapter 6. Separability and Trans-separability 1. Separability of Function Spaces 2. Trans-separability of Function Spaces 3. Notes and Comments

101 102 107 114

Chapter 7. Weak Approximation in Function Spaces 1. Weak Approximation in (Cb (X, E), β) 2. Weak Approximation in CVo (X, E) 3. Notes and Comments

115 116 120 126

Chapter 8. The Riesz Representation type Theorems 1. Vector-valued Measures and Integration 2. Integral Representation Theorems 3. Notes and Comments

127 128 135 140

Chapter 9. Weighted Composition Operators 1. Multiplication Operators on CVo (X, E) 2. Weighted Composition Operators on CVo (X, E) 3. Compact Weighted Composition Operators 4. Notes and Comments

141 142 152 159 166

Chapter 10. The General Strict Topology 1. Strict Topology on Topological Modules 2. Essentiality of Modules of Vector-valued Functions 3. Topological Modules of Homomorphisms 4. Notes and Comments

167 168 177 180 187

Chapter 11. Mean Value Theorem and Almost Periodicity 1. The Mean Value Theorem in TVSs 2. Almost Periodic Functions with Values in a TVS 3. Notes and Comments

189 190 194 203

Chapter 12. Non-Archimedean Function Spaces 1. Compact-open Topology on C(X, E) 2. Strict Topology on Cb (X, E) 3. Maximal Ideals in NA Function Algebras 4. Notes and Comments

205 206 214 222 227

Appendix A. Topology and Functional Analysis 1. Topological Spaces 2. Topological Vector Spaces 3. Spaces of Continuous Linear Mappings 4. Shrinkable Neighborhoods in a TVS

229 230 243 261 268

CONTENTS

5. 6. 7. 8. 9.

Non-Archimedean Functional Analysis Topological Algebras and Modules Measure Theory Uniform Spaces and Topological Groups Notes and Comments

v

272 275 296 300 310

Appendix. Bibliography

313

Appendix B. List of Symbols

329

Appendix C. Index

333

vi

CONTENTS

Introduction This monograph is devoted to the study of linear topological spaces of continuous bounded vector-valued functions endowed with the uniform, strict, compact-open and weighted topologies. In the past four decades, several major results on vector-valued function spaces have been extended to the non-locally setting. These include generalized versions of some classical results such as the Stone-Weierstrass theorem, the ArzelaAscoli theorem, and the Riesz representation theorem. These also include maximal ideal spaces of function algebras, separability and transseparability, weak approximation, composition operators, general strict topology on topological modules, the mean value theorem and almost periodicity for vector-valued functions, and non-Archimedean function spaces. Our main objective is to present recent developments in these areas. Several examples and counter-examples are included in the text. Background material on Topology and Functional Analysis is included in the Appendix. Also, an up-to-date bibliography is included to assist research in further studies. AMS Subject Classification. [2000] Primary 46E10, 46E40; Secondary 46A10, 28A25, 46H25, 47B33

Preface The purpose of this monograph is to present an up-to-date study of linear topological spaces of continuous bounded vector-valued functions endowed with the uniform, strict, compact-open and weighted topologies. In the past four decades, several new results have had been obtained in this direction. The main topics include generalized versions of some classical results such as the Stone-Weierstrass theorem, the Arzela-Ascoli theorem, and the Riesz representation theorem. These also include the study of maximal ideal spaces of function algebras, separability and transseparability, weak approximation, composition operators, general strict topology on topological modules, the mean value theorem and almost periodicity for vector-valued functions, and non-Archimedean function spaces. Our main objective is to present recent developments in the nonlocally convex setting, that is, in topological vector spaces, not necessarily locally convex. So it covers both the cases: (a) locally convex spaces, (b) topological vector spaces which are not locally convex. We mention that ”non-convexity” has also been the main theme in the monographs by L. Waelbroeck (1971), N. Adasch, B. Ernst and D. Keim (1978), N.J. Kalton, N.T. Peck and J.W. Roberts (1984), S. Rolewicz (1985) and A. Bayoumi (2003); see also the papers by W. Robertson (1958), V. Klee (1960a, 1960b) and J. Kakol (1985, 1987, 1990, 1992). Our exposition relies mostly on the original papers (published during 1972-2012) with some additional clarifications. We have tried to make the text easily readable and as self-contained as possible. The only prerequisites for reading the book are topology, functional analysis and measure theory of the undergraduate level (e.g. the material given in H.L. Royden’s book: Real Analysis). For this purpose, we have included background material on Topology and Functional Analysis (such as topological spaces, topological vector spaces, non-Archimedean functional analysis, topological algebras and measure theory) in Appendix A. Also, an up-to-date bibliography is included to assist research in further studies. The monograph is intended basically a monograph for researchers working in vector-valued function spaces. However, those working in this and vii

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related fields may conveniently choose some topics for a one or two semester course work. In fact, the author has been using portions of the text in his lectures to M.Sc./M.Phil. students as a one semester course before initiating their research in a relevant field of function spaces. To introduce the material in its historical perspective, consider a topological space X and a topological vector space E over K (= R or C). Let C(X, E) (resp. Cb (X, E)) be the vector space of all continuous (resp. continuous and bounded) E-valued functions on X; if E = K, these spaces are simply denoted by C(X) and Cb (X). If X is compact, then C(X, E) = Cb (X, E) and C(X) = Cb (X); in this case the uniform topology (i.e. the sup norm topology) u is the appropriate one to study on C(X). After the appearance of M.H. Stone’s paper of 1937, the space (C(X), u) in the case of compact X has been an intensively studied mathematical object. Its interest arises in part from its rich structure: under the uniform topology u, C(X) is a Banach space; under pointwise multiplication, it is an algebra; under the natural ordering, C(X, R) is a lattice (see, e.g., Kakutani (1941), Kaplansky (1947b), Hewitt (1948), Myers (1950), M. and S. Krein (1940), etc.). If X is not compact, the uniform topology u on C(X) is not well-defined since C(X) may contain some unbounded functions. However, in this case, the compact-open topology k is the most useful topology to be considered on C(X) and on C(X, E) with E even a topological space. It has been studied by several authors over the past sixty years and significant contributions have been made in this field, among others, by Fox (1945), Arens (1946), Myers (1946), Hewitt (1948), Warner (1958), Wheeler (1976), McCoy (1980); see also the monographs by Semadeni (1971), Schmets (1983), McCoy and Ntantu (1988), Arkhangel’skii (1992) and Tkachuk (2011). If X is again non-compact and we consider the spaces Cb (X) and Cb (X, E), then both u and k topologies are well-defined and we have k 6 u. In 1958, R.C. Buck introduced the notion of strict topology β on Cb (X, E) in the case of X locally compact and E a locally convex space. The problems discussed in the Buck’s paper (1958) are: (1) Relationship between the β, k and u topologies on Cb (X, E) (e.g. k 6 β 6 u with k = u iff X is compact; k = β on u-bounded sets); (2) Completeness of Cb (X, E), β); (3) Stone-Weierstrass theorems for (Cb (X), β) and Cb (X, En ), β); (4) Characterization of maximal β-closed ideals in Cb (X); (5) Identification of the β-dual of Cb (X) with the space M(Bo(X)) of regular Boreal measures on X, via the integral representations; (6) The open problem whether or not (Cb (X), β) a Mackey space.

PREFACE

ix

After the appearance of the Buck’s paper, a large number of papers have appeared in the literature concerned with extending Buck’s results to more general cases or studying further properties of β, and also with obtaining some variants of β. In particular Todd (1965) and Wells (1965)], independently, established the Stone-Weierstrass theorem for (Cb (X), β). Todd (1965) also characterized the maximal β-closed Cb (X)-submodules of Cb (X, E) while Wells (1965) identified the β-dual of Cb (X, E) with a certain space M(X, E ∗ ) of E ∗ -valued measures on X. Conway (1967) and LeCam (1957), independently, proved that, if X is locally compact and paracompact, (Cb (X, E), β) is a Mackey space but it is not so in general. Collins and Dorroh (1968) proved that (Cb (X), β) has the approximation property. Dorroh (1969) also showed that β is the finest locally convex topology on Cb (X) which agrees with k on ubounded sets. The next major development in the study of β topology on Cb (X) has been its extension to the case of a completely regular space X, given, independently, by Van Rooij (1967), Giles (1971), Fremlin, Garling and Haydon (1972), Gulick (1972), Hoffman-Jφregensen (1972), and Sentilles (1972). In fact, Fremlin, Garling and Haydon (1972) and, independently, Sentilles (1972) introduced on Cb (X) three types of strict topologies: the substrict topology βo , the strict topology β and the superstrict topology β1 with β being equivalent to the strict topology ‘β’ of Buck (1958) in the case of X a locally compact space. They also identified the topological duals of (Cb (X), βo ), (Cb (X), β) and (Cb (X), β1 ) with the spaces Mt (X), Mτ (X) and Mσ (X) of tight, τ -additive and σ-additive measures, respectively, on X (see LeCam (1957), Varadarajan (1965) and Wheeler (1983) for detail of these spaces of measures). Subsequently, several authors have further explored the properties of βo , β and β1 for both Cb (X) and Cb (X, E), where E is a normed space or a locally convex space; see, e.g. Mosiman and Wheeler (1972), Wheeler (1973), Summers (1972), Haydon (1976), Cooper (1971), Choo (1979), Fontenot (1974), Katsaras (1975), Khurana (1978a, 1978b), Morishita and Khan (1997), Zafarani (1986, 1988), and also the survey papers by Hirschfeld (1978), Collins (1976) and Wheeler (1983), and monographs by Cooper (1978), Prolla (1977), Schmets (1983), Singh and Manhas (1993). In 1967, Nachbin studied in detail the more general notion of weighted topology ωV on certain subspaces CVb (X) and CVo (X) of C(X) and β, k, and u were shown to be the special cases of ωV for suitable choices of V consisting of non-negative upper-semicontinuous functions on X. Further work on ωV has been done by Summers (1969, 1971), Prolla

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(1971a, 1971b), Bierstedt (1973, 1975), and others. Singh and Summers (1988) and Singh and Manhas (1991, 1992) made an extensive study of composition and multiplication operators, respectively on CVb (X, E). Another useful topology on Cb (X) is the σ-compact-open topology σ which was introduced by Gulick (1972) and further studied by Gulick and Schmets (1972). The strict and related topologies have also been studied on not necessarily ‘function spaces’. For instance, Wiwegar (1961) and Cooper (1971, 1978) defined it in the form of ‘mixed topology’ on a normed space; Busby (1968) considered it on the double centralizer (or multiplier) algebra Md (A) of Banach algebra A; Sentilles and Taylor (1969) defined it on a Banach A-module; Ruess (1977) studied it on an arbitrary locally convex space (see the survey paper by Collins (1976)). In all the above mentioned investigations about Cb (X, E), E has been assumed to be the scalar field or a locally convex space. The case of E a general topological vector space has been first considered by Shuchat in (1972a, 1972b) where he established several useful approximation results for (Cb (X, E), u) and characterized the dual of (Cb (X, E), u), with X a compact space and without assuming the local convexity of E (see also Bierstedt (1973) and Wealbroeck (1971, 1973)). In 1979, Khan defined the β and other related topologies on Cb (X, E), where X is a Hausdorff space and E any Hausdorff topological vector space, and showed that β has almost all the properties of the “strict topology” studied by the above authors. Further work in this direction has been done in later years by Khan (1980 through 2011), Khan-Rowlands (1981, 1991), Katsaras (1981, 1983), Kalton (1983), Nawrocki (1985, 1989), Abel (1987, 2004), Prolla (1993b), Khan-Thaheem (1997, 2002), Manhas-Singh (1998), Khan-Mohammad-Thaheem (1999, 2005), KhanOubbi (2005), Katsaras-Khan-Khan (2011), Katsaras (2011) and others. The purpose of this monograph is to present some results of these authors regarding the strict, weighted, and related topologies on Cb (X, E) and CVo (X, E) in the non-locally convex setting. We also study the strict and related topologies on any topological modules. Several examples and counter-examples are included in the text. . The monograph consists of twelve chapters which are organized as follows. In Chapter 1, we introduce and study the strict topology β (and the related ones such as the uniform topology u, compact-open topology k, σcompact-open topology σ) and also the more general notion of weighted topology ωV on the function spaces Cb (X, E) (resp. CVb (X, E)) with E

PREFACE

xi

not necessarily a locally convex topological vector space. Here, we are mainly concerned with their linear topological properties. Chapter 2 deals with completeness and quasi-completeness of function spaces under the above mentioned topologies. In Chapter 3, we establish the Arzela-Ascoli type theorems which characterize the β-compact and ωV -precompact subsets of Cb (X, E) and CVb (X, E), respectively. Chapter 4 contains various generalizations of the famous Stone-Weierstrass theorem and related results. It is perhaps the most important one from the point of view of its applications. Chapter 5 deals with characterization of maximal and closed ideals in (Cb (X, E), β) and (C(X, E), k), E a topological algebra, and also their maximal ideal spaces ∆((Cb (X, E), β)) and ∆((C(X, E), k)). In Chapter 6, we present various versions of the M and S. Krein type theorem for separability and trans-separability in the β, k, σ, and u topologies. Chapter 7 deals with the problem of weak approximation in (Cb (X, E), β) and (CVo (X, E), ωV ), using some basic terminology of Measure Theory. In Chapter 8, we first study the integral of functions in Cb (X, E) with respect to certain E ∗ -valued measures, and then obtain the Riesz representation type theorems for the dual of function spaces under various topologies. In Chapter 9, we study necessary and sufficient conditions for certain maps Mθ , Mψ , Cϕ and Wπ,ϕ to be the multiplication or composition operators on the weighted spaces CVo (X, E) and CVb (X, E). We also include characterizations of compact weighted composition operators and compact multiplication operators. We have treated the non-commutative analogue of the strict and related topologies in Chapter 10. In fact, we introduce and study the general strict and related topologies on a topological left A-module Y . We also study properties of the topological module HomA (A, Y ) of continuous homomorphisms. Chapter 11 consists of some results on the mean value theorem and almost periodicity for vector-valued functions. In Chapter 12, we have considered the study of compact-open and strict topologies on non-Archimedian function spaces. Finally, in the Appendix A, we have included basic definitions and results on Topological Spaces, Topological Vector Spaces, Non-Archimedean

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Functional Analysis, Topological Algebras, Topological Modules, Measure Theory and Uniform Spaces. Most of their proofs (with the exception of some selected results) are omitted since they are available in any standard book on these topics. At the end of each chapter, the section ”Notes and Comments” contains references to the literature and gives some further information on the results in the text. Some references are also included in the text to clarify the source of information. The bibliography lists some papers that are not specifically referred but can be useful for supplementary material. We have treated the cases of strict topology and weighted topology in separate sections in most of the chapters. Any one interested only in more familiar results on uniform, strict and compact-open topologies can easily skip the material on the more abstract notion of weighted topology. Liaqat Ali Khan

Acknowledgements I wish to express my sincere gratitude to Professors S. M. Khaleelulla, Mati Abel and Taqdir Husain for reading the first draft of manuscript and offering several useful comments for improvement. I also wish to thank Keith Rowlands (the supervisor of my Ph.D. thesis of 1977), A.V. Arkhangel’skii, Surjit Singh Khurana, and my colleagues Abdul Latif, Naseer Shahzad and Lakhdar Meziani for discussions and consultation during the preparation of this monograph. I thank Mr. M. Sharaz for his skillful typing of the material. Finally, the author wishes to acknowledge with thanks the Deanship of Scientific Research, King Abdulaziz University, Jeddah, for their technical and financial support.

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The Strict and Weighted Topologies In this chapter, we study the uniform, strict, compact-open, σ-compactopen, weighted and other related topologies on Cb (X, E) or CVb (X, E) and their linear topological properties in the non-locally convex setting. Throughout this monograph, we shall assume, unless stated otherwise, that X is a completely regular Hausdorff space and E a non-trivial Hausdorff topological vector space (in short, a TVS) over the field K = (R or C), and we let W denote a base of open balanced neighborhoods of 0 in E. Most of the undefined terminology can be found in the Appendix A (A.1-A.8).

1

2

1. THE STRICT AND WEIGHTED TOPOLOGIES

1. Strict and related Topologies on Cb (X, E) The notion of the strict topology β on Cb (X, E) was first introduced by Buck [Buc58] in 1958 in the case of X a locally compact space and E a locally convex TVS. Since then a large number of papers have appeared in the literature concerned with extending the results contained in Buck’s paper; see the Preface and the list of references. In this section, a general method for defining linear topologies on Cb (X, E), called the S-topologies, is considered first. Using the notion of an S-topology, the uniform topology u, strict topology β, compact-open topology k, σ-compact-open topology σ, countable-open-topology σo , and pointwise topology p are introduced. We include here some results from [Kh79, KR91, Kat81], where E is a Hausdorff TVS but not necessarily locally convex. Definition. A function f : X → E is said to bounded if f (X) is a bounded subset of E, i.e. for each W ∈ W, there exists an r > 0 such that f (X) ⊆ λW for all λ ∈ K with |λ| ≥ r. f : X → E is said to vanish at infinity if, for each W ∈ W, the set FW = {x ∈ X : f (x) ∈ / W} is relatively compact in X; in particular, a function ϕ : X → K is said to vanish at infinity if, for each ε > 0, the set Fε = {x ∈ X : |ϕ(x)| ≥ ε} is relatively compact in X. The support of f : X → E is defined as the closure of the set {x ∈ X : f (x) 6= 0} in X and we write supp(f ) = cl − {x ∈ X : f (x) 6= 0}. Let C(X, E) be the vector space of all continuous E-valued functions on X and Cb (X, E) (resp. Co (X, E), Coo (X, E)) the subspace of C(X, E) consisting of those functions which are bounded (resp. vanish at infinity, have compact support}. When E = K (= R or C), then these spaces will be denoted by C(X), Cb (X), Co (X) and Coo (X). Let B(X) denote the see of all bounded K-valued functions on X, and let Bo (X), (resp. Boo (X), Bσ (X), Bσo (X), Bp (X)) denote the subset of B(X) consisting of those functions which vanish at infinity (resp. have compact, σ-compact, countable, finite support). We shall denote by B(X)⊗E the vector space spanned by the set of all functions of the form ϕ ⊗ a, where ϕ ∈ B(X), a ∈ E, (ϕ ⊗ a)(x) = ϕ(x)a (x ∈ X). F (X, E) = E X denotes the vector space of all functions f : X → E.

1. STRICT AND RELATED TOPOLOGIES ON Cb (X, E)

3

Since we shall frequently use the notion of a function vanishing at infinity, the following three lemmas are useful in this regard. Lemma 1.1.1 Let ϕ : X → K be a function. Then: (a) ϕ vanishes at infinity iff, for each ε > 0, there exists a compact set Kε ⊆ X such that | ϕ(x) |< ε for all x ∈ X\Kε . (b) If ϕ is non-negative upper semi-continuous (in particular, continuous), then ϕ vanishes at infinity iff for every ε > 0, the set Fε = {t ∈ X : |ϕ(t)| ≥ ε} is compact iff for every ε > 0, there exists a compact set Kε ⊆ X such that | ϕ(x) |< ε for all x ∈ X\Kε . Proof. (a) Suppose ϕ vanishes at infinity, and let ε > 0. Then the set Fε = {x ∈ X : |f (x)| ≥ ε} is relatively compact in X. Now the set Kε = Fε is compact. If x ∈ X\Kε , then x ∈ X\Fε ; hence | ϕ(x) |< ε. Conversely, let ε > 0, and let Kε be the compact set such that | ϕ(x) |< ε for all x ∈ X\Kε . If x ∈ / Kε , | ϕ(x) |< ε and so clearly x ∈ / Fε . Hence Fε ⊆ Kε . Now Kε , being a compact subset of the Hausdorff space X, is closed. Therefore Fε ⊆ Kε = Kε . Then Fε , being a closed subset of compact set Kε , is compact. (b) If ϕ is a non-negative upper semi-continuous, then the set {x ∈ X : ϕ(x) < ε} = {x ∈ X : |ϕ(x)| < ε} is open, or that Fε is closed. Then the result follows from (a). Analogously, we also obtain: Lemma 1.1.2 Let f : X → E be a function. Then: (a) f vanishes at infinity iff, for each W ∈ W, there exists a compact set KW ⊆ X such that f (x) ∈ W for all x ∈ X\KW . (b) If f is continuous, then f vanishes at infinity iff, for each W ∈ W, the set FW = {x ∈ X : f (x) ∈ / W } is compact iff for each W ∈ W, there exists a compact set KW ⊆ X such that f (x) ∈ W for all x ∈ X\KW . Lemma 1.1.3. Let f : X → E be a function and suppose ϕf ∈ B(X, E) for each ϕ ∈ Bo (X). Then f ∈ B(X, E). Proof. Suppose f is not bounded. Then there exist a sequence {xn } ⊆ X, and a W ∈ W such that f (xn ) ∈ / n2 W . Define ϕ : X → R by ϕ(x) = 1/n if x = xn , and ϕ(x) = 0 if x 6= xn

(n = 1, 2, ...).

4


Then ϕ ∈ Bo (X). [Clearly ϕ is bounded. To show that ϕ vanishes at infinity, let ε > 0. Choose N ≥ 1 such that N1 < ε. Then K = {x1 , ..., xN } is compact and if x ∈ / K = {x1 , ..., xN }, then 1 1 ϕ(x) ≤ < < ε. N +1 N Alternatively, 1 ≤ ϕ(x) ≤ 1} = {x1 , ..., xN }, {x ∈ X : ϕ(x) ≥ ε} = {x ∈ X : N which, being a finite set, is compact in X.] Now, clearly 1 ϕ(xn )f (xn ) = f (xn ) ∈ / nW for all n = 1, 2, .... n This implies that ϕf is not bounded, which contradicts our hypothesis. Thus f ∈ B(X, E). Remarks. (1) The above result need not hold if X is not Hausdorff. For example, let X = N with the topology τ = {N} ∪ {An : n ∈ N}, where An = {i : i ∈ N, i < n} for every n ∈ N. This is T0 but not T1 (hence also not Hausdorff). In this case, the only closed compact subset of X is ∅, so the only relatively compact set is ∅. Consequently, the only member of Bo (N) is the constant function with value 0. Now consider the TVS E = R and the identity function f : N → R, f (n) = n, n ∈ N. Then ϕf is bounded for every ϕ ∈ Bo (N), because there is only the zero function ϕ to consider; but f is not bounded. (2) Clearly, a finite subset A of a topological space X is always compact; but need not be closed. So it need not be relatively compact (e.g. if X is neither T1 nor regular). Notation. For any ϕ ∈ B(X) and W ∈ W, we define N(ϕ, W ) = {f ∈ Cb (X, E) : ϕ(x)f (x) ∈ W for all x ∈ X}. If A ⊆ X and W ∈ W, then we write N(A, W ) to mean N(A, W ) := N(χA , W ) = {f ∈ Cb (X, E) : f (A) ⊆ W }, where χA is the characteristic function of A. Lemma 1.1.4. Let ϕ ∈ B(X) and W ∈ W. Then (a) If V ∈ W with V ⊆ W, then N(ϕ, V ) ⊆ N(ϕ, W ). (b) If U, V ∈ W with U + V ⊆ W, then N(ϕ, U) + N(ϕ, V ) ⊆ N(ϕ, W ). (c) For any λ(6= 0) ∈ K, N(ϕ, λW ) = λN(ϕ, W ).


5

(d) N(ϕ, W ) is absorbing. (e) If W is balanced (resp. convex), then N(ϕ, W ) is also balanced (resp. convex). Proof. (a) Let f ∈ N(ϕ, V ). Then, for any x ∈ X, ϕ(x)f (x) ∈ V ⊆ W and so f ∈ N(ϕ, W ). (b) Let f ∈ N(ϕ, U) and g ∈ N(ϕ, V ). Then, for any x ∈ X, ϕ(x)f (x) ∈ U and ϕ(x)g(x) ∈ V, and so [ϕ(f + g)](x) = ϕ(x)f (x) + ϕ(x)g(x) ∈ U + V ⊆ W ; that is, f + g ∈ N(ϕ, W ). Hence N(ϕ, V ) + N(ϕ, V ) ⊆ N(ϕ, W ). (c) If λ(6= 0) ∈ K, N(ϕ, λW ) = {f ∈ Cb (X, E) : ϕ(x)f (x) ∈ λW for all x ∈ X} 1 = λ{ f ∈ Cb (X, E) : ϕ(x)f (x) ∈ λW for all x ∈ X} λ 1 1 = λ{ f ∈ Cb (X, E) : ϕ(x)( f )(x) ∈ W for all x ∈ X} λ λ = λ{g ∈ Cb (X, E) : ϕ(x)g(x) ∈ W for all x ∈ X} = λN(ϕ, W ). (d) Let f ∈ Cb (X, E). Since (ϕf )(X) is bounded in E and W is absorbing, there exists r > 0 such that (ϕf )(X) ⊆ λW for all | λ |≥ r. Hence, by (c), ϕf ∈ N(ϕ, λW ) = λN(ϕ, W ) for all | λ |≥ r, showing that N(ϕ, W ) is absorbing. (e)(i) Suppose W is balanced, and let λ ∈ K with |λ| ≤ 1. Then λW ⊆ W and so, by (a) and (c), λN(ϕ, W ) = N(ϕ, λW ) ⊆ N(ϕ, W ). Hence N(ϕ, W ) is balanced. (ii) Suppose W is convex, and let t ∈ R with 0 < t < 1 and f, g ∈ N(ϕ, W ). Then tW + (1 − t)W ⊆ W , and so by (b) and (c), tN(ϕ, W ) + (1 − t)N(ϕ, W ) = N(ϕ, tW ) + N(ϕ, (1 − t)W ) ⊆ N(ϕ, tW + (1 − t)W ) ⊆ N(ϕ, W ). Hence N(ϕ, W ) is convex. We now describe a general method of defining linear topologies on Cb (X, E).

6


Definition. Let S be any subset of B(X). We define the S-topology on Cb (X, E) to be the linear topology which has a subbase of neighborhoods of 0 consisting of all sets of the form N(ϕ, W ), where ϕ ∈ S and W ∈ W. Definition. [Gil71] Let ϕ, ϕ1 ∈ B(X). Then ϕ1 is said to dominate ϕ if there exists a t > 0 such that |ϕ(x)| ≤ t |ϕ1 (x)| for all x ∈ X. Lemma 1.1.5. [Gil71, Kh79] Let S and S1 be two subsets of B(X). If each element of S is dominated by an element of S1 , then the S-topology on Cb (X, E) is weaker than the S1 -topology. Proof. Let N be any Sneighborhood of 0 in Cb (X, E), and suppose N ⊇

n \

N(ϕi , Wi ),

i=1

where ϕ1 , ..., ϕn ∈ S and W1 , ..., Wn ∈ W. For each ϕi (i = 1, ..., n), choose a ti > 0 and a ψi ∈ S1 such that |ϕi (x)| ≤ ti |ψi (x)| for all x ∈ X. Let N1 =

n \

1 N(ψi , Wi ), t i=1

where t = max{t1 , ..., tn }. Then N1 is an S1 -neighborhood of 0 in Cb (X, E). Further N1 ⊆ N . [Let g ∈ N1 . Then, for any x ∈ X and 1 ≤ i ≤ n, 1 ψi (x)g(x) ∈ Wi ; t since |ϕi (x)| ≤ ti |ψi (x)| ≤ t|ψi (x)| and Wi is balanced, ϕi (x)g(x) ∈ tψi (x)Wi ⊆ Wi . Therefore g ∈ ∩ni=1 N(ϕi , Wi ) ⊆ N .] Hence N is an S1 -neighborhood of 0 in Cb (X, E). Definition. Using the notion of an S-topology, we now introduce the strict topology, the σ-compact-open topology, and other related topologies on Cb (X, E), as follows.


S B(X) Bo (X) Boo (X) Bσ (X) Bσo (X) Bp (X)

7

S-Topology uniform topology u strict Topology β compact-open topology k σ-compact-open topology σ countable-open-topology σo pointwise topology p

Remark 1.1.6. It easily follows from Lemma 1.1.5 that: (1) The u-topology on Cb (X, E) is the same as the {1}-topology, where 1 ∈ B(X) is the function identically 1 on X. (2) The topology k (resp. σ, σo , p) on Cb (X, E) is the linear topology which has a subbase of neighborhoods of 0 consisting of all sets of the form N(K, W ), where K is any compact (resp. σ-compact, countable, finite) subset of X and W ∈ W. (3) Since Bp (X) ⊆ Boo (X) ⊆ Bo (X) ⊆ B(X) and Bp (X) ⊆ Bσo (X) ⊆ Bσ (X) ⊆ B(X), it is easy to verify that p 6 k 6 β 6 u and p 6 σo 6 σ 6 u. Further, note that if X is compact, then k = u; if X is discrete, then p = k (since every compact set in a discrete space is finite). We mention that the topologies p and k can also be considered on the larger space C(X, E). The following lemma gives us a convenient form for a base of neighborhoods of 0 in Cb (X, E) for each of the topologies defined above. Lemma 1.1.7. [Kh79] Let S denote any one of the sets B(X), Bo (X), Boo (X), Bσ (X), Bσo (X), or Bp (X). Then the S-topology on Cb (X, E) has a base of neighborhoods of 0 consisting of all sets of the form N(ϕ, W ), where ϕ ∈ S with 0 ≤ ϕ ≤ 1 and W ∈ W. Proof. Let N be any Sneighborhood of 0, and suppose m \ N ⊇ N(ϕi , Wi ), i=1

where ϕ1 , ..., ϕm ∈ S and W1 , ..., Wm ∈ W. Let t = max {kϕi k}. If t > 0, choose a W ∈ W with tW ⊆ ∩m i=1 Wi . Define |ϕi (x)| ϕ(x) = max (x ∈ X). 1≤i≤m t

1≤i≤m

8


Then ϕ ∈ S, 0 ≤ ϕ ≤ 1. Further, N(ϕ, W ) ⊆ N . [Let g ∈ N(ϕ, W ). Since, for each x ∈ X and 1 ≤ i ≤ m, |ϕi (x)| ≤ tϕ(x) and ϕ(x)g(x) ∈ W, we have ϕi (x)g(x) ∈ tW ⊆ Wi ; that is, g ∈ ∩m i=1 N(ϕi , Wi ) ⊆ N .] If t = 0, then each ϕi = 0 and so N(ϕi , Wi ) = {f ∈ Cb (X, E) : 0f (x) ∈ W for all x ∈ X} = Cb (X, E). Hence N = Cb (X, E), and so, if we take ϕo = 0, we have N ⊇ N(ϕo , W ) for any W in W. Thus the Sneighborhoods of 0 have a base of the required form. We first study some basic properties of the topology β and its relation with the topologies k and u. (For convenience, the properties of the topologies σo and σ will be given separately in a later theorem.) Theorem 1.1.8. [Buc58, Kh79] (a) p 6 k 6 β 6 u. (b) If X is completely regular, then: (i) u = β iff X is compact. (ii) β = k iff every σ-compact subset of X is relatively compact. (c) u and β have the same bounded sets in Cb (X, E). (d) β = k on u-bounded subsets of Cb (X, E). (e) A sequence {fn } in Cb (X, E) is β-convergent iff it is u-bounded and k-convergent. Proof. (a) Since every element of Bo (X) is dominated by 1 ∈ B(X), it follows from Lemma 1.1.5 that the Bo (X)-topology is weaker than the {1}-topology, that is, β 6 u. Similarly, since Boo (X) ⊆ Bo (X), we have k 6 β. (b) (i) Suppose u 6 β. Then, by Lemma 1.1.7, for any W ∈ W, there exist a ϕ ∈ Bo (X) with 0 ≤ ϕ ≤ 1 and a V ∈ W such that N(ϕ, V ) ⊆ N(1, W ). If E\W 6= ∅, let c ∈ E\W and choose t > 0 such that c ∈ tV . If X is not compact, then X\F 6= ∅ for every compact set F in X. Since ϕ ∈ B0 (X), the set K = cl − {x ∈ X : ϕ(x) > 1/t} is compact in X. Let xo ∈ X\K. By complete regularity, there exists a ψ ∈ Cb (X) such that 0 ≤ ψ ≤ 1, ψ(xo ) = 1 and ψ(K) = 0. Let g = ψ ⊗ c. Then g ∈ N(ϕ, V ). [Let x ∈ X. If x ∈ K, then ψ(x) = 0 and so ϕ(x)g(x) = ϕ(x)ψ(x)c = 0 ∈ V ;


9

if x ∈ / K, then tϕ(x) < 1 and so ϕ(x)g(x) = ϕ(x)ψ(x)c ∈ ϕ(x)ψ(x)tV ⊆ V .] But g ∈ / N(1, W ) (since g(xo ) = ψ(xo )c = c ∈ / W ), which is a contradiction. If W = E, choose a W0 in W such that W0 ⊆ E and then argue as above with W0 replacing W . Thus X is compact. Conversely, if X is compact, then k = u and so, by (a), β = u. (b) (ii) If every σ-compact subset of X is relatively compact, then it is easy to show that β 6 k. Conversely, let β 6 k, and suppose that there is a set G = ∪∞ n=1 Kn (Kn compact in X) which is not relatively compact. Then, for each compact set F in X, G\F 6= ∅. Let ∞ X 1 ϕ= χK . 2n n n=1

Then ϕ ∈ Bo (X). [Clearly ϕ is bounded. To show that ϕ vanishes at N Kn is infinity, let ε > 0. Choose N ≥ 1 such that N1 < ε. Now K = Un=1 compact. If x ∈ / K, then χKn (x) = 0 for all n = 1, 2, ..., N and so X 1 X 1 1 1 (x) ≤ χ = N < < ε.] ϕ(x) = K n n n 2 2 2 N n≥N +1 n≥N +1 Clearly, ϕ = 0 outside of G. Since β 6 k, for any balanced W ∈ W, there exist a compact set K in X and a V ∈ W such that N(K, V ) ⊆ N(ϕ, W ). If E\W 6= ∅, let d ∈ E\W , and yo ∈ G\K. Choose a ψ ∈ Cb (X) with 0 ≤ ψ ≤ (1/ϕ(yo)), ψ(yo ) = 1/ϕ(yo), and ψ(K) = 0. Let h = ψ ⊗ d. Then, since h(K) = 0 ⊆ V, h ∈ N(K, V ) but, since ϕ(yo )h(yo ) = d ∈ / W, we have h ∈ / N(ϕ, W ), a contradiction. If W = E, choose a Wo in W such that Wo ⊆ E and then argue as above with Wo replacing W . (c) Since β 6 u, it follows that every u-bounded subset of Cb (X, E) is β-bounded. Conversely, suppose there is a set A ⊆ Cb (X, E) which is β-bounded but not u-bounded. Then there exist a sequence {fn } ⊆ A, {xn } ⊆ X, and a W ∈ W such that fn (xn ) ∈ / n2 W. Let ϕ(x) = 1/n if x = xn , and ϕ(x) = 0 if x 6= xn (n = 1, 2, ...). Then, as seen in the proof of Lemma 1.1.3, ϕ ∈ Bo (X) but ϕ(xn )fn (xn ) ∈ / nW ;

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that is, {fn } and hence A is not β-bounded. This contradiction proves the result. (d) Let A be a u-bounded subset of Cb (X, E). Since k 6 β, it is sufficient to show that the identity mapping i : (A, k) → (A, β) is continuous. Suppose f ∈ A and let {fα } be a net in A such that k- lim fα = f. α

Let ϕ ∈ Bo (X) with 0 ≤ ϕ ≤ 1, and let W ∈ W. We show that there exists an index αo such that ϕ(x)(fα (x) − f (x)) ∈ W for all x ∈ X and α > αo . Choose a balanced V ∈ W such that V + V ⊆ W . Since A is u-bounded, choose a t > 1 such that g(x) ∈ tV for all x ∈ X and g ∈ A. Since ϕ ∈ Bo (X), there exists a compact set K in X such that 1 for x ∈ / K. t Let ϕ1 = ϕχK . Then ϕ1 ∈ Boo (X) and so, since k-limα fα = f , there exists an index αo such that ϕ(x)
αo . Let y be any point of X. If y ∈ K, then ϕ1 = ϕ, so that ϕ(y)(fα (y) − f (y)) ∈ V ⊆ W for all α > αo . If y ∈ / K, then tϕ(y) < 1 ϕ(y)(fα (y) − f (y)) ∈ ϕ(y)tV − ϕ(y)tV ⊆ V + V ⊆ W for all α. Thus β-limα fα = f and so i : (A, k) → (A, β) is continuous; that is, β|A ⊆ k|A, as required. (e) Suppose {fn } is β-convergent in Cb (X, E). Since a convergent sequence in a TVS is always bounded, it follows that {fn } is β-bounded and hence u-bounded by (c). Moreover, since k 6 β, {fn } is k-convergent. Conversely, suppose {fn } is u-bounded in Cb (X, E) and k- lim fn = n→∞

f , where f is in Cb (X, E). The {f, fn : n = 1, 2..} is contained in a u-bounded subset, say A, of Cb (X, E). By (d) β|A = k|A, and so, it follows that β- lim fn = f . n→∞ The following remarks contain some counterexamples related to the above results.


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Remark 1. Since k 6 β, every β-bounded set is k-bounded. But the converse is not true, in general; Hirschfeld ([Hir74], p.20) has given the following example of a k-bounded set which is not β-bounded. Example. Let X = N, so that Cb (N) = ℓ∞ . Define ϕn = nχ[n,∞); that is, n if m > n ϕn (m) = (m ∈ X). 0 if m < n

Let K ⊆ X = N be any compact set. Then K is finite, there exists an integer no such that ϕn |K = 0 for all n > no . Since K is arbitrary, k-lim n

ϕn = 0. So {ϕn } is k-bounded. On the other hand, supn {kϕn k} = ∞ and so {ϕn } is not u-bounded or equivalently {ϕn } is not β-bounded. Remark 2. It is easy to see that, if a net {fα } in Cb (X, E) is ubounded and k-convergent, then {fα } is β-convergent. However, the converse is not true; Gulick ([Gu72], p.166) has given an example of a β-convergent net which is not u-bounded as follows. Example. Let X = Ω, the space of ordinals less than the first uncountable ordinal ω, with order topology. Let Ωo ⊆ Ω the collection of non-limit ordinals. Let fλ = nλ χ{λ} where λ is nλ (0 < nλ < ∞) greater β

than a limit ordinal. The fλ → 0. But {fλ : λ ∈ Ω\Ωo } is not u-bounded. We now discuss the relation of β with the Buck’s strict topology [Buc58]. Let β ′ denote the Co (X)-topology, which is the Buck’s original strict topology, on Cb (X, E). Theorem 1.1.9. [Kh79] If X is locally compact, β ′ = β on Cb (X, E). Proof. It is clear that β ′ 6 β, since Co (X) ⊆ Bo (X). Now, let ϕ ∈ Bo (X). By Lemma 1.1.5, it is sufficient to show that there exists a function ψ in Co (X) which dominates ϕ. For each n, the set 1 } 2n has a compact closure Kn say in X. Since X is locally compact, there exist function ψn ∈ Coo (X) such that {x ∈ X : |ϕ(x)| >

0 ≤ ψn ≤ 1, ψn (Kn ) = 1, and supp (ψn ) ⊆ Kn . Let ψ =

∞ P

n=1

1 ψ . 2n n

Clearly Kn ⊆ Kn+1 and ψ ∈ Co (X). To show |ϕ| ≤ ψ

on X, let y ∈ X. If y ∈

∞ S

n=1

Kn , then y ∈ KN +1 \KN for some N > 1, so

12


that ψ(y) ≥ If y ∈ /

∞ S

X 1 1 = N > |ϕ(y)|. n 2 2 n>N

Kn , then ϕ(y) = 0 but ψ(y) > 0. Thus |ϕ(y)| ≤ ψ(y).

n=1

Consequently β 6 β ′ .

Remark. If X is not locally compact, then Co (X, E) may be the trivial vector space {0}. Counter-example. Let X = Q, the space of rationals and E = R. Then Co (Q, R) = {0}, as follows. Suppose ϕ(6= 0) ∈ Co (Q, R). Choose y ∈ Q with ϕ(y) 6= 0. We may suppose ϕ(y) > 0. Choose a, b > 0 such that a < ϕ(y) < b. Then the set U = {x ∈ Q : a < ϕ(x) < b} = ϕ−1 (a, b) is non-empty (since y ∈ U) and open (since ϕ is continuous). Since ϕ ∈ Co (Q, R), the set V = {x ∈ Q : |ϕ(x)| > a} = {x ∈ Q : ϕ(x) > a or ϕ(x) < −a} is relatively compact. Now, for any x ∈ U, ϕ(x) > a and so x ∈ V ; that is, U ⊆ V . Then it follows that U is compact. Hence every point in U has a compact neighborhood. Upon translating U along Q, we arrive at the contradiction that Q is locally compact. Theorem 1.1.10. [Buc58, Kh79] Let E be any TVS. If X is locally compact, then: (a) Coo (X, E) is β-dense in Cb (X, E). (b) Coo (X, E) is k-dense in C(X, E). (c) Coo (X, E) is u-dense in Co (X, E). (d) Co (X, E) is u-closed in Cb (X, E). Conversely, if E is a non-trivial TVS, then each of the condition (a), (b) implies that X is locally compact; if, in addition, Co (X, E) 6= {0}, then (c) also implies that X is locally compact. Proof. Suppose X is locally compact. (a) Let f ∈ Cb (X, E). Let ϕ ∈ Bo (X), 0 ≤ ϕ ≤ 1, and W ∈ W. Let K ⊆ X be a compact set such that ϕ(x)f (x) ∈ W for x ∈ / K. Sine X is locally compact, choose a function ψ ∈ Coo (X) such that 0 ≤ ψ ≤ 1 and ψ(K) = 1. Let g = ψf . Then g ∈ Coo (X) and ϕ(x)[g(x) − f (x)] = ϕ(x)[ψ(x) − 1]f (x) =0∈W if x ∈ K ∈ (ψ(x) − 1)W ⊆ W if x ∈ / K.


13

Thus g − f ∈ N(ϕ, W ), and so f belongs to the β-closure of Coo (X, E); that is, Coo (X, E) is β-dense in Cb (X, E). (b) Let f ∈ C(X, E). Let K be a compact subset of X. Choose a ψ ∈ Coo (X) such that ψ = 1 on K. Then ψf ∈ Coo (X, E) and (ψf )(x) − f (x) = 0 for all x ∈ K. In particular, for any W ∈ W, ψf − f ∈ N(K, W ), and so f belongs to the k-closure of Coo (X, E) Thus Coo (X, E) is k-dense in C(X, E). (c) Let f ∈ Co (X, E). To show that f belongs to the u-closure of Coo (X, E), let W ∈ W be balanced. Now K = {x ∈ X : f (x) ∈ / W } is compact. Choose ψ ∈ Coo (X) such that 0 ≤ ψ ≤ 1 and ψ = 1 on K. Then clearly g = ψf ∈ Coo (X, E). Let x ∈ X. If x ∈ K, then f (x) − g(x) = f (x) − f (x) = 0 ∈ W. If x ∈ X\K, then f (x) − g(x) = [ψ(x) − 1]f (x) ∈ [ψ(x) − 1]W ⊆ W. Hence g − f ∈ N(X, W ). Thus Coo (X, E) is u-dense in Co (X, E). u (d) Let f ∈ F (X, E) with f ∈ Co (X, E) . Then there exists a seu quence {fn } ⊆ Co (X, E) such that fn −→ f . We need to show that f ∈ Co (X, E). f is continuous on X, as follows Let xo ∈ X and W ∈ W. We show that there exists a neighborhood G of xo in X such that, f (y)−f (xo ) ∈ W for all y ∈ G.Choose a balanced V ∈ W such that V + V + V ⊆ W . u Since fn −→ f, there exists n0 ≥ 1 such that fn (x) − f (x) ∈ V for all n ≥ n0 and all x ∈ X.

(1)

Since fn0 is continuous at xo , there exists a neighborhood G of xo in X such that fn0 (y) − fn0 (xo ) ∈ V for all y ∈ G. (2) Then, for any y ∈ G, using (1) and (2), f (y) − f (xo ) = (f (y) − fn0 (y)) + (fn0 (y) − fn0 (xo )) + (fn0 (xo ) − f (xo )) ∈ V + V + V ⊆ W. Hence f is continuous on X. Thus f ∈ Cb (X, E) We now need to verify that f vanishes at infinity. Let W ∈ W. u Choose a balanced V ∈ W such that V + V ⊆ W . Since fn −→ f , there exists k0 ≥ 1 such that fn (x) − f (x) ∈ V for all n ≥ N0 and x ∈ X.

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Since fk0 ∈ C0 (X, E), there exists a compact set K ⊆ X such that fk0 (x) ∈ V for all x ∈ X\K. Then, for any x ∈ X\K, f (x) = (f (x) − fk0 (x)) + fk0 (x) ∈ V + V ⊆ W . Therefore f ∈ Co (X, E). Thus Co (X, E) is u-closed in Cb (X, E). Conversely, suppose E is a non-trivial TVS. (i) Suppose Coo (X, E) is β-dense in Cb (X, E) but that X is not locally compact. There there exists a y ∈ X which has no compact neighborhood. Consequently g(y) = 0 for all g ∈ Coo (X, E). Now, let W ∈ W be balanced. Since E is non-trivial, we may assume W 6= E and choose a ∈ E\W. Consider the non-zero constant function h : X → E function given by h(x) = a for all x ∈ X. Then clearly h ∈ Cb (X, E). Now, N({y}, W ) = {f ∈ Cb (X, E) : f (y) ∈ W } is a pneighborhood of 0 in Cb (X, E) and clearly, for any g ∈ Coo (X, E), h(y) − g(y) = a − 0 ∈ / W; i.e., h − g ∈ / N({y}, W ). Therefore h does not belong to the p-closure of Coo (X, E); since p ≤ β, h does not belong to the β-closure of Coo (X, E). Therefore Coo (X, E) is not β-dense in Co (X, E), a contradiction. Thus X is locally compact. (ii) Suppose Coo (X, E) is k-dense in C(X, E) but that X is not locally compact. There there exists a y ∈ X which has no compact neighborhood. Consequently g(y) = 0 for all g ∈ Coo (X, E). It follows (as in (i)) that, if h is any non-zero constant function in C(X, E), then h does not belong to the p-closure, and hence to the k-closure of Coo (X, E) (since p ≤ k); that is Coo (X, E) is not k-dense in C(X, E). (iii) Suppose Co (X, E) 6= {0} and that Coo (X, E) is u-dense in Co (X, E) but that X is not locally compact. There there exists a y ∈ X which has no compact neighborhood. Consequently g(y) = 0 for all g ∈ Coo (X, E). It follows (as in (i)) that, if h is any non-zero function in Co (X, E), then h does not belong to the p-closure, and hence not to the u-closure of Coo (X, E); that is Coo (X, E) is not u-dense in Co (X, E). If X is not assumed to be locally compact, then a result related to the above theorem can be obtained by imposing a stronger condition on E, as follows. Theorem 1.1.11. [Kh95] Suppose E is a locally bounded TVS. Then Cb (X, E) is k-dense in C(X, E).


15

Proof. Let f ∈ C(X, E), and let K be a compact subset of X and W ∈ W. By hypothesis, we can choose a bounded set V ∈ W. Let S be a closed shrinkable neighborhood of 0 in E with S ⊆ V (see Section A.4). The Minkowski functional ρS of S is continuous and positively homogeneous, and so, for any r > 0, the function hr : E → E defined by a if a ∈ rS hr (a) = r a if a ∈ / rS ρS (a)

is continuous with hr (E) ⊆ rS. This implies that the function hr ◦ f ∈ Cb (X, E). [Clearly, hr ◦ f is continuous on X. To show that hr ◦ f is bounded on X, let U ∈ W. Choose λ > 0 with V ⊆ λU. Then hr (f (X)) ⊆ hr (E)) ⊆ rS ⊆ rV ⊆ rλU.] Choose t ≥ 1 with f (K) ⊆ tS. Then ht ◦ f ∈ Cb (X, E). Further, for any x ∈ K, f (x) ∈ tS and so ht (f (x)) − f (x) = f (x) − f (x) = 0 ∈ W. Thus Cb (X, E) is k-dense in C(X, E). We next consider the properties of σ and σo topologies on Cb (X, E).

Lemma 1.1.12. [KR91] Suppose that A, B ⊆ X and that V, W ∈ W are such that N(A, V ) ⊆ N(B, W ). Then: (i) If W 6= E, then B ⊆ A. (ii) If B is non-empty, then V ⊆ W . Proof. (i) Suppose B * A, and let x ∈ B\A. Since X is completely regular, there exists a ϕ in Cb (X) such that ϕ(x) = 1 and ϕ(A) = {0}. Then clearly supp (ϕ) ∩ A = ∅. Choose a ∈ E\W and let f = ϕ ⊗ a. Then f (A) = {0} ⊆ V , and so f ∈ N(A, V ); but f (x) = a ∈ / W and so f∈ / N(B, W ). This contradiction implies that B ⊆ A. (ii) Suppose V W . Choose b ∈ V \W . and let g = χX ⊗ b. If A = ∅, then g(A) = ∅ ⊆ V ; if A 6= ∅, then g(A) = {b} ⊆ V . Hence g ∈ N(A, V ). Since B 6= ∅, g(B) = {b} * W and so g ∈ / N(B, W ), a contradiction. Theorem 1.1.13. [KR91] Let p, k, β, σ, u and σo be the topologies on Cb (X, E) as defined above. Then (a) p 6 k 6 β 6 σ 6 u and p 6 σo 6 σ. (b) σ = u iff X = A for some σ-compact subset A of X. (c) k = σ and σ = β iff every σ-compact subset of X is relatively compact. (d) σo = u iff X is separable. (e) σo 6 k iff every countable subset of X is relatively compact.

16


(f ) σo , σ and u have the same bounded sets in Cb (X, E). Proof. (a) Clearly, p 6 σo 6 σ 6 u (as noted in Remark 1.1.6) and, by Theorem 1.1.8(a), p 6 k 6 β 6 u. Hence, we only need to verify that β 6 σ. Let ϕ ∈ Bo (X), with 0 ≤ ϕ ≤ 1, and W ∈ W. Then there exists a sequence {Kn } of compact sets in X such that ϕ(x) < 1/n for x ∈ / Kn , which implies that ϕ = 0 outside of the set A = ∪∞ n=1 Kn and N(A, W ) ⊆ N(ϕ, W ), giving β 6 σ. (b) If X = A for some σ-compact subset A of X, then it is clear that u 6 σ. Conversely, suppose that u 6 σ and let W ∈ W, with W 6= E. Then there exist a σ-compact subset A of X and a V ∈ W such that N(A, V ) ⊆ N(X, W ). By Lemma 1.1.12, X = A, as required. (c) If every σ-compact subset of X is relatively compact, then clearly σ 6 k. Conversely, suppose that σ 6 k and let A be any σ-compact subset of X. If W ∈ W and W 6= E, then there exist a compact set B and a V ∈ W such that N(B, V ) ⊆ N(A, W ). By Lemma 1.1.12, A ⊆ B, which implies that A is relatively compact. The second part of the assertion follows from part (b) and Theorem 1.1.8(b)(ii). (d) Since a separable set is the closure of a countable set, the σo topology is equivalent to the topology of uniform convergence on the separable subsets of X. Thus, if X is countable, it is clear that u 6 σo . Conversely, if u 6 σo , then, for any W ∈ W, W 6= E, there exist a separable set A and a V ∈ W such that N(A, V ) ⊆ N(X, W ). By Lemma 1.1.12, X = A and so X is separable. (e) This is similar to that of part (c). (f) Since σo 6 u, every u-bounded set is σo -bounded. Conversely, suppose that there exists a σo -bounded set H which is not u-bounded. Then there exist sequences {fn } ⊆ H, {xn } ⊆ X and a W ∈ W such that fn {xn } ∈ / nW.


17

Let A = {xn }. Then H is not absorbed by N(A, W ), which is a contradiction. Recall that a TVS E is called ultrabornological if every bounded linear map from E into any TVS F is continuous; E is called ultrabarrelled if every closed linear map from E into any complete metrizable TVS F is continuous; E is called quasi-ultrabarrelled if every bounded linear map from E into any complete metrizable TVS F is continuous. Every metrizable TVS is ultrabornological and every complete metrizable TVS is ultrabarrelled. Further, every ultrabornological and every ultrabarrelled TVS is quasi-ultrabarrelled. Theorem 1.1.14. [Buc58, Kh79, KR91] Let t denote any one of the topologies σo , β or σ on Cb (X, E). (a) If t is ultrabornological, then t = u on Cb (X, E). (b) If E is metrizable, then the following are equivalent: (i) t = u. (ii) t is metrizable. (iii) t is ultrabornological. Proof. (a) The identity map i : (Cb (X, E), t) → (Cb (X, E), u) is linear and bounded and so, since t is ultrabornological, i is continuous. Hence u 6 t. (b) If E is metrizable, then (Cb (X, E), u) is also metrizable. Therefore (i) ⇒ (ii) ⇒ (iii) is clear and (iii) ⇒ (i) follows from part (a). Remarks. (1) If E is a complete metrizable TVS, so is (Cb (X, E), u) and therefore in Theorem 1.1.14 (b), we have (i) ⇒ (iv) ⇒ (v), where: (iv) t is ultrabarrelled (v) t is quasi-ultrabarrelled. (2) In view of Theorem 1.1.14(a), the topologies σo , β and σ are not ultrabornological or metrizable (even in the case of real-valued function). In fact, if X is not pseudocompact, then β 6= u on Cb (X). The following theorem which gives an alternate form of β-neighborhoods of 0 in Cb (X, E) is due to Katsaras [Kat81]. Theorem 1.1.15. [Kat81] β has a base of neighborhoods of 0 in Cb (X, E) consisting of all sets of the form N = N (Kn , an , W ) =

n \

{f ∈ Cb (X, E) : f (Kn ) ⊆ an W },

(∗)

i=1

where {Kn } is a sequence of compact subsets of X, {an } a sequence of positive numbers with an → ∞ and W ∈ W.

18


Proof. Let N be of the form (∗) with W balanced. Then ϕ := sup{ n≥1

1 χK } ∈ Bo (X). an n

Moreover, N(ϕ, W ) ⊆ N ,as follows. Let f ∈ N(ϕ, W ). Then 1 χK (x)f (x) = ϕ(x)f (x) ∈ W for all x ∈ X, an n and so f (Kn ) ⊆ an W for all n > 1. Therefore f ∈ N (Kn , an , W ). Hence N is a β-neighborhood of 0 in Cb (X, E). Conversely, let No be a β-neighborhood of 0 in Cb (X, E). There exists a balanced neighborhood W of 0 in E and ϕ ∈ Bo (X) with 0 ≤ ϕ ≤ 1 such that N1 = N(ϕ, W ) ⊆ No . For n > 1, there exists a compact subset Kn of X such that {x ∈ X : g(x) >

1 } ⊆ Kn . n

Set ∞ \ n n N2 = N (Kn , , W ) = {f ∈ Cb (X, E) : f (Kn ) ⊆ W }. 2 2 n=1

Then N2 ⊆ N1 , as follows. Let f ∈ N2 . Then f (Kn ) ⊆ n2 W for all n > 1. Let x ∈ X. If ϕ(x) 6= 0, then there exists n > 2 such that 1 1 < ϕ(x) < and so x ∈ Kn \Kn+1. n n−1 Then, since n < 2(n − 1), f (x) ∈ Therefore, ϕ(x)f (x) ∈ required.

n 1 2 W and ϕ(x) < < . 2 n−1 n

n W 2(n−1)

⊆ W, and so f ∈ N1 . Thus N2 ⊆ No , as

Corollary 1.1.16. [Kat81] If N is a β-neighborhood of 0 in Cb (X, E), then there exists a balanced W ∈ W with the following property: For each d > 0, there exist a compact set K ⊆ X and a positive number δ > 0 such that {f ∈ Cb (X, E) : f (X) ⊆ dW, f (K) ⊆ δW } ⊆ N .


19

Proof. Since N is a β-neighborhood of 0 in Cb (X, E), there exist ϕ ∈ Bo (X), 0 ≤ ϕ ≤ 1, and W ∈ W such that N(ϕ, W ) ⊆ N . For each n > 1, let 1 Kn = {x ∈ X : ϕ(x) > }. n Then each Kn is compact, Kn−1 ⊆ Kn , ϕ = 1 on K1 , 1 1 ≤ ϕ(x) < for x ∈ Kn \Kn−1 (n ≥ 2), n n−1 ∞ S and ϕ = 0 outside Kn . Let now d > 0 be given. Choose an integer n=1

N such that N > 2d. Let K =

N S

n=1

Kn = KN , δ = 21 . We now verify that

{f ∈ Cb (X, E) : f (X) ⊆ dW, f (K) ⊆ δW } ⊆ N(ϕ, W ). Let f ∈ Cb (X, E) with f (X) ⊆ dW, f (K) ⊆ δW . Let x ⊆ X. Suppose x ∈ K. If x ∈ K1 , ϕ(x) = 1, and so 1 ϕ(x)f (x) ∈ δW = W ⊆ W. 2 If x ∈ Kj \Kj−1 (2 ≤ j ≤ N), and so 1 W ⊆ W. ϕ(x)f (x) ∈ 2(j − 1) ∞ S Now, let x ∈ [ Kn ]\KN . Then x ∈ Kp \Kp−1 for some p > N + 1. Then n=1

ϕ(x)
r}) is open. Clearly, ϕ : X → R is continuous iff it is both upper and lower semicontinuous. Further, the characteristic function χA of a subset A of X is upper (resp. lower) semicontinuous iff A is closed (resp. open). Every upper (resp. lower) semicontinuous function assumes its supremum (resp. infimum) on a compact set. In particular, every non-negative upper semicontinuous function on a compact set is bounded. Definition. A Nachbin family V on X is a set of non-negative upper semicontinuous functions on X such that, given u, v ∈ V and λ > 0, there exists a w ∈ V such that λu, λv ≤ w (pointwise). Let CVb (X, E) (resp. CVo (X, E)) denote the subspace consisting of those f ∈ C(X, E) such that vf is bounded (resp. vanishes at infinity) for all v ∈ V . If U and V are two Nachbin families on X and, for every u ∈ U, there is a v ∈ V such that u ≤ v (pointwise), then we write U ≤ V . If V is a Nachbin family on X such that, for every x ∈ X, there is a v ∈ V with v(x) 6= 0, then we write V > 0. Lemma 1.2.1. [Pro71b, Kh85b] Let U and V be two Nachbin families on X with U ≤ V . Then CVb (X, E) ⊆ CUb (X, E) and CVo (X, E) ⊆ CUo (X, E). Proof. The first inclusion is immediate from the definition. Let f ∈ CVo (X, E), and let u ∈ U and G ∈ W. Choose a v ∈ V with u ≤ v. There exists a compact set K ⊆ X such that v(x)f (x) ∈ G for all x ∈ / K. Let y ∈ / K ∩ supp(u). If y ∈ / K, then u(y)f (y) ∈ G. If y ∈ / supp(u), then u(y) = 0, and so again u(y)f (y) ∈ G. Hence f ∈ CUo (X, E). We now list some important classes of Nachbin families on X. For any subset A of X, χA will denote its characteristic function. (1) Kf+ (X) = {λχF : λ > 0 and F ⊆ X, F finite}. (2) Kc+ (X) = {λχK : λ > 0 and F ⊆ X, K compact}.

2. WEIGHTED TOPOLOGY ON CVb (X, E)

21

+ (3) Soo (X), the set of all non-negative upper semi-continuous functions on X which have compact support, (4) So+ (X), the set of all non-negative upper semi-continuous functions on X which vanish at infinity. (5) K + (X), the set of all non-negative constant functions on X. Clearly, + Kf+ (X) 6 Kc+ (X) 6 Soo (X) 6 So+ (X) 6 K + (X).

If V is a Nachbin family on X with V > 0, then it is easily seen that Kf+ (X) 6 V. Lemma 1.2.2. [Pro71b, Kh85b] If V = So+ (X), then CVb (X, E) = CVo (X, E) = Cb (X, E). Proof. Clearly, CVo (X, E) ⊆ CVb (X, E) and Cb (X, E) ⊆ CVb (X, E). Since every v ∈ So+ (X) is bounded on a compact set, it follows that CVb (X, E) ⊆ CVo (X, E). To show that CVb (X, E) ⊆ Cb (X, E), let f ∈ CVb (X, E). Suppose f is not bounded. Then there exist a G ∈ W and a sequence {xn } ⊆ X such that f (xn ) ∈ / 22n G for all n > 1. P∞ 1 Let ϕ = n=1 2n χ{xn } . Then ϕ ∈ So+ (X) and, for any n > 1, ϕ(xn )f (xn ) ∈ / 2n G.

This implies that ϕf is not bounded, a contradiction. Hence f ∈ Cb (X, E). We also note that: (1) If V = K + (X), then CVb (X, E) = Cb (X, E) and CVo (X, E) = Co (X, E). + (2) If V = Kc+ (X) or V = Soo (X), then

CVb (X, E) = CVo (X, E) = C(X, E). (3) If V =

Kf+ (X),

then also CVb (X, E) = CVo (X, E) = C(X, E).

Definition. [Pro71b, Kh85b] Let V be a Nachbin family on X. The weighted topology ωV on CVb (X, E) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form N(v, G) = {f ∈ CVb (X, E) : (vf )(X) ⊆ G}, where v ∈ V and G ∈ W.

22


The assumption V > 0 implies that p ≤ ωV . Recall that p ≤ k on C(X, E) and k ≤ β ≤ u on Cb (X, E). Theorem 1.2.3. [Pro71b, Kh85b] If V = So+ (X), then ωV = β, the strict topology on CVb (X, E) = Cb (X, E). Proof. It is clear that ωV 6 β, since So+ (X) ⊆ Bo (X). Now, let ϕ ∈ Bo (X). By Lemma 1.1.5, it is sufficient to show that there exists a function ψ in So+ (X) which dominates ϕ. For each n, the set 1 {x ∈ X : |ϕ(x)| > n } 2 P 1 has compact closure, K say, in X. Let ψ = ∞ n=1 2n χKn . Clearly

Kn ⊆ Kn+1 and ψ ∈ So+ (X). ∞ S To show |ϕ| ≤ ψ on X, let y ∈ X. If y ∈ Kn , then y ∈ Kn+1 \KN for n=1

some N > 1, so that

ψ(y) ≥ If y ∈ /

∞ S

X 1 1 = N > |ϕ(y)|. n 2 2 n>N

Kn , then ϕ(y) = 0 but ψ(y) > 0. Thus |ϕ(y)| ≤ ψ(y).

n=1

Consequently β 6 ωV , as required.

Remark. Similarly, as above, we note that: (1) If V = K + (X), then ωV is the uniform topology u on CVb (X, E) = Cb (X, E). (2) If V = Kc+ (X), then ωV is the compact-open topology k on CVb (X, E) = C(X, E). (3) If V = Kf+ (X), then ωV is the pointwise topology p on CVb (X, E) = C(X, E). Lemma 1.2.4. Let U and V be Nachbin families on X with U 6 V . Then ωU 6 ωV on CVb (X, E). Proof. This is clear. Theorem 1.2.5. [Pro71b, Kh85b] Let V be a Nachbin family on X such that χc (X) 6 V and all v ∈ V vanish at infinity. Then (a) k 6 ωV 6 β 6 u on Cb (X, E). (b) k and ωV coincide on u-bounded subsets of Cb (X, E). Proof. (a) This follows immediately from Lemma 1.2.4.

2. WEIGHTED TOPOLOGY ON CVb (X, E)

23

(b) Let A be a u-bounded subset of Cb (X, E), and let f belong to the k-closure of A in Cb (X, E). Take any v ∈ V and G ∈ W. Choose a balanced H ∈ W with H + H ⊆ G. Choose r > 0 such that f (X) ⊆ rH for all g ∈ A. Let 1 K = {x ∈ X : v(x) > } and s = sup{v(x) : x ∈ K}. r There exists an h ∈ A such that (h − f )(K) ⊆ 1s H. Then, for any y ∈ X, v(y) 1s H ⊆ G if y ∈ K v(y)[h(y) − f (y)] ∈ v(y)(rH − rH) ⊆ G if y ∈ / K;

that is, h − f ∈ N(v, G). Hence f belongs to the ωV -closure of A, and so ωV 6 k on A. The following example shows that CVb (X, E) may be trivial for relatively nice X. Example. Let X = Q, the set of all rationals with the natural topology. This is of course a metrizable space. Consider on X the Nachbin family V = C + (X) consisting of all non-negative continuous functions. We claim that CVb (X, E) is reduced to {0} for every non-trivial TVS E. [Indeed, assume that, for a given TVS E, CVb (X, E) 6= {0}, and let f (6= 0) ∈ CVb (X; E). Then f (xo ) 6= 0 for some xo ∈ X. Since E is a Hausdorff TVS, there exists some shrinkable neighborhood G ∈ W so that ρG (f (xo )) 6= 0. With no loss of generality, we assume that ρG (f (xo )) = 1. Since ρG ◦ f : X → R is continuous at xo , taking ε = 12 , there exists δ > 0 such that if |x − xo | < δ, 1 |ρG (f (x) − ρG (f (xo )| < ; 2 1 1 in particular, ρG (f (x)) > ρG (f (xo ) − 2 = 2 . For an irrational t ∈ P with |t − xo | < δ, the function vt : X → R+ given by 1 , x ∈ X, vt (x) = |t − x| belongs to V = C + (X) and then, since f ∈ CVb (X; E), vt must verify sup{vt (x)ρG (f (x)) : x ∈ X} < +∞. But sup{vt (x)ρG (f (x)) : x ∈ X} ≥ vt (xo )ρG (f (xo )) = a contradiction.]

1 → ∞ as t → xo , |t − xo |

24


3. Notes and Comments Section 1.1. A general method for defining linear topologies on Cb (X, E), called the S-topologies, is considered first. Using the notion of an S-topology, the uniform topology u, strict Topology β, compactopen topology k, σ-compact-open topology σ, countable-open-topology σo , and pointwise topology p are introduced. See also ([KN63], Section 8, p. 68-82). The topologies k and p on both C(X) and C(X, Y ), Y a uniform space, have been extensively studied over the past sixty years and significant contributions have been made in this field, among others, by Fox [Fox45], Arens [Are46a], Myers [My46], Hewitt [Hew48], Warner [war58], Wheeler [Whe76], Schmets [Schm83] and McCoy [Mc80]. In recent years, the Cp -theory has been actively investigated by several mathematicians (see the monographs by Arkhangel’skii [Ark92] and Tkachuk [Tka11], and their references). The notion of the strict topology β on Cb (X, E) was first introduced by Buck [Buc58] in 1958 in the case of X a locally compact space and E a locally convex TVS. (In fact, this was also considered earlier in 1952 by Buck himself [Buc52] on Cb (X) for the special case of X a group; it was there called the ”strict topology” because of its resemblance to a topology used by Beurling [Beu45].) Since then a large number of papers have appeared in the literature concerned with extending the results contained in Buck’s paper; see the Preface and the list of references. Another useful topology on Cb (X, E) is the σ-compact-open topology σ which was introduced by Gulick [Gu72] and studied further by Gulick and Schmets [GuSc72]. Most of these investigations have been concerned with generalizing the space X and taking E to be the scalar field or a locally convex space. This section includes some results from [Kh79, KR91, [Kat81], where E is a Hausdorff TVS but not necessarily locally convex. Section 1.2. This section is devoted to the study of weighted topology ωV on the weighted function space CVb (X, E), as given in [Kh85, Kh87]. In particular, the relation of weighted topology with the uniform, strict, compact-open and pointwise topologies is also given. The fundamental work on weighted spaces had been mainly done by Nachbin [Nac65, Nac67]. Further investigations have been made by Summers [SumW69, SumW72], Prolla [Pro71, Pro72, Pro77], Bierstedt [Bie73] and other authors (see the monograph [Pro77] and survey article [Col76] for more references). Most of these investigations have been

3. NOTES AND COMMENTS

25

concerned with generalizing the space X and taking E to be the scalar field or a locally convex space. Later, Khan [Kh85, Kh87], Nawrocki [Naw89], Prolla [Pro93], Khan and co-authors [KT97, KT02, KMT05, KO05] and Manhas and Singh [MS98] have studied various aspects of the strict, weighted and other related topologies, where E is a Hausdorff TVS. These results are given in later chapters.

CHAPTER 2

Completeness in Function Spaces In this chapter, we study various completeness properties of the uniform strict, weighted and other related topologies.

27

28

2. COMPLETENESS IN FUNCTION SPACES

1. Completeness in the uniform topology In this section, we consider completeness of the spaces (B(X, E), u), (Cb (X, E), u) and (Co (X, E), u). Let X be a topological space and E a TVS with a base W of neighborhoods of 0. Let B(X, E) denotes the vector space of all bounded functions f : X → E. The uniform topology u on B(X, E) is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form N(X, W ) = {f ∈ B(X, E) : f (X) ⊆ W }, where W over W. If E = (E, d) is metrizable, then (B(X, E), u) and (Cb (X, E), u) are also metrizable with respect to the metric given by: ρ(f, g) = sup d(f (x), f (x)), f, g ∈ B(X, E) or Cb (X, E). x∈X

Theorem 2.1.1. [KN63] Let X be a topological space and E a complete metrizable TVS. Then: (a) (B(X, E), u) is complete, hence an F-space. (b) (Cb (X, E), u) is complete, hence an F-space. (c) If X is locally compact, then (C0 (X, E), u) is complete, hence an F-space. Proof. (a) Let {fn } be a u-Cauchy sequence in Cb (X, E). Since p ≤ u, {fn } is a p-Cauchy net in C(X, E); that is, for each x ∈ X, {fn (x)} is a Cauchy net in E. Since E is complete, for each x ∈ X, f (x) = u limn→∞ fn (x) exists in E. We need to show that fn −→ f and that f is continuous and bounded. u Now, fn −→ f , as follows. Let W ∈ W be closed. Since {fn } is u-Cauchy, there exists n0 ∈ I such that fn (x) − fm (x) ∈ W for all n, m ≥ n0 and x ∈ X. Fix any n ≥ n0 . Since W is closed and, for each x ∈ X, fm (x) −→ f (x), we have fn (x) − f (x) ∈ W for x ∈ X. Thus fn (x) − f (x) ∈ W for all n ≥ n0 and all x ∈ X; i.e. fn − f ∈ N(X, W ) for all n ≥ n0 . u

This proves that fn −→ f .

1. COMPLETENESS IN THE UNIFORM TOPOLOGY

29

Finally, f is bounded as follows. Let W ∈ W. Choose a balanced u V ∈ W such that V + V ⊆ W . Since fn −→ f , there exists an n0 ≥ 1 such that fn (x) − f (x) ∈ V for all n ≥ n0 and x ∈ X. Since fn0 is bounded, we can choose a constant r > 1 such that fn0 (x) ⊆ rV for all x ∈ X. Then, for any x ∈ X, f (x) = [f (x) − fn0 (x)] + fn0 (x) ∈ V + rV ⊆ rV + rV = r(V + V ) ⊆ rW . Hence f is bounded on X. Then (Cb (X, E), u) is complete. (b) Let {fn : n ∈ I} be a u-Cauchy net in Cb (X, E). Since p ≤ u, {fn } is a p-Cauchy net in Cb (X, E); that is, for each x ∈ X, {fn (x)} is a Cauchy net in E. Since E is complete, for each x ∈ X, f (x) = limn fn (x) u exists in E. We need to show that fn −→ f and that f is continuous and bounded. u As shown in part (a), f is bounded and that fn −→ f . Therefore, we only need to show that f is continuous on X. Let x0 ∈ X and W ∈ W. We show that there exists a neighborhood G of x0 in X such that f (y) − f (x0 ) ∈ W for all y ∈ G. u

Choose a balanced V ∈ W such that V + V + V ⊆ W . Since fn −→ f, there exists and n0 ∈ I such that fn (x) − f (x) ∈ V for all n ≥ n0 and all x ∈ X.

(1)

Since fn0 is continuous at x0 , there exists a neighborhood G of x0 in X such that fn0 (y) − fn0 (x0 ) ∈ V for all y ∈ G. (2) Then, for any y ∈ G, using (1) and (2), f (y) − f (x0 ) = (f (y) − fn0 (y)) + (fn0 (y) − fn0 (x0 )) + (fn0 (x0 ) − f (x0 )) ∈ V + V + V ⊆ W. Hence f is continuous on X. Thus f ∈ Cb (X, E), and so (Cb (X, E), u) is complete. (c) Let {fn } be a u-Cauchy sequence in C0 (X, E). Since p ≤ u, {fn } is a p-Cauchy sequence in C0 (X, E); that is, for each x ∈ X, {fn (x)} is a Cauchy sequence in E. Since E is complete, for each x ∈ X, f (x) = limn→∞ fn (x) exists in E. Then as in part (a), f is continuous and that u fn −→ f . We now need to verify that f ∈ C0 (X, E). Let W ∈ W.

30

2. COMPLETENESS IN FUNCTION SPACES u

Choose a balanced V ∈ W such that V + V ⊆ W . Since fn −→ f , there exists N0 ≥ 1 such that fn (x) − f (x) ∈ V for all n ≥ N0 and x ∈ X. Since fk0 ∈ C0 (X, E), there exists a compact set K ⊆ X such that fk0 (x) ∈ V for all x ∈ X\K. Then, for any x ∈ X\K, f (x) = [f (x) − fk0 (x)] + fk0 (x) ∈ V + V ⊆ W . Therefore f ∈ C0 (X, E). Thus (C0 (X, E), u) is complete.

Theorem 2.1.2. Let X be a locally compact Hausdorff space and E any TVS. Then: (a) (C0 (X, E), u) is closed in Cb (X, E). (b) If, in addition, E is complete, then (C0 (X, E), u) is also complete. u Proof. (a) Let f ∈ F (X, E) with f ∈ C0 (X, E) . Then there exists u a sequence {fn } ⊆ F (X, E) such that fn −→ f . We need to show that f ∈ C0 (X, E). To show that f is continuous on X. Let x0 ∈ X and W ∈ W. We show that there exists a neighborhood G of x0 in X such that f (y) − f (x0 ) ∈ W for all y ∈ G. u

Choose a balanced V ∈ W such that V + V + V ⊆ W . Since fn −→ f, there exists and n0 ∈ I such that fn (x) − f (x) ∈ V for all n ≥ n0 and all x ∈ X.

(1)

Since fn0 is continuous at x0 , there exists a neighborhood G of x0 in X such that fn0 (y) − fn0 (x0 ) ∈ V for all y ∈ G. (2) Then, for any y ∈ G, using (1) and (2), f (y) − f (x0 ) = (f (y) − fn0 (y)) + (fn0 (y) − fn0 (x0 )) + (fn0 (x0 ) − f (x0 )) ∈ V + V + V ⊆ W. Hence f is continuous on X. Thus f ∈ Cb (X, E) Next, f is bounded on X, as follows. Let W ∈ W. Choose a balanced u V ∈ W such that V + V ⊆ W . Since fn −→ f , there exists an n0 ≥ 1 such that fn (x) − f (x) ∈ V for all n ≥ n0 and x ∈ X. Since fn0 is bounded, we can choose a constant r > 1 such that fn0 (x) ⊆ rV for all x ∈ X.

1. COMPLETENESS IN THE UNIFORM TOPOLOGY

31

Then, for any x ∈ X, f (x) = [f (x) − fn0 (x)] + fn0 (x) ∈ V + rV ⊆ rV + rV = r(V + V ) ⊆ rW . Hence f is bounded on X. We now need to verify that f vanishes at infinity. Let W ∈ W. u Choose a balanced V ∈ W such that V + V ⊆ W . Since fn −→ f , there exists N0 ≥ 1 such that fn (x) − f (x) ∈ V for all n ≥ N0 and x ∈ X. Since fk0 ∈ C0 (X, E), there exists a compact set K ⊆ X such that fk0 (x) ∈ V for all x ∈ X\K. Then, for any x ∈ X\K, f (x) = [f (x) − fk0 (x)] + fk0 (x) ∈ V + V ⊆ W . Therefore f ∈ C0 (X, E). Thus (C0 (X, E), u) is closed in Cb (X, E). (b) If E is complete, then, by Theorem 2.1.1 (a), (Cb (X, E), u) is complete. Hence, C0 (X, E), being a closed subset of the complete space C0 (X, E), is also complete. Note. The above theorem gives an alternate proof of Theorem 2.1.1(c). Recall that a TVS F is called quasi-complete (or boundedly complete ) if every bounded closed subset H of F is complete. Clearly completeness implies quasi-completeness. Further, every closed subset of a quasicomplete TVS is quasi-complete. Theorem 2.1.3. Let E be a quasi-complete TVS. Then: (a) (Cb (X, E), u) and (B(X, E), u) and are quasi-complete. (b) If X is locally compact, then (Co (X, E), u) is quasi-complete. Proof. (a) Let H be a closed and bounded set on (Cb (X, E), u), and let {fα } be a u-Cauchy net in H. Clearly, for each x ∈ X, {f (x) : f ∈ H} is a bounded set in E; in particular, for each x ∈ X, {fα (x)} is a bounded net in E. Further, since p ≤ u, for each x ∈ X, {fα (x)} is a Cauchy net in E. Since E is quasi-complete, for each x ∈ X, f (x) := limα fα (x) exists in E, and let f (x) := lim fα (x), x ∈ X. α

32

2. COMPLETENESS IN FUNCTION SPACES u

fα −→ f : Let W ∈ W be closed. Since {fα } is u-Cauchy, there exists αo ∈ I such that fα (x) − fγ (x) ∈ W for all α, γ ≥ αo and x ∈ X. Fix any α ≥ αo . Since W is closed and, for each x ∈ X, fγ (x) −→ f (x), we have fα (x) − f (x) ∈ W for all x ∈ X. u Thus fα − f ∈ N(X, W ) for all α ≥ αo . This proves that fα −→ f . Finally, since {fα } ⊆ H and H is u-closed, f ∈ H. This shows that H is complete; hence (Cb (X, E), u) is quasi-complete. By the same argument, (B(X, E), u) is also quasi-complete. (b) It is similar to part (a).

2. COMPLETENESS IN THE STRICT TOPOLOGY

33

2. Completeness in the Strict Topology We consider completeness, sequential completeness and quasi-completeness of the β, σ, σo on Cb (X, E) and ωV topologies on CVb (X, E) and of k topology on C(X, E). Recall that a topological space (X, τ ) is called: (i) a k-space if, for any U ⊆ X, U is closed in X whenever U ∩ K is closed in K for each compact K ⊆ X; (ii) a kR -space if a function f : X → R is continuous on X whenever f | K is continuous for every compact subset K of X; Every locally compact space and every first countable (in particular, metric) space is a k-space. If X is a k-space and Y any topological space, then a function f : X → Y is continuous on X iff f | K is continuous for every compact subset K of X; hence every k-space is a kR -space. Theorem 2.2.1. [Buck58, Kh79] Let X be a k-space and E a complete TVS. Then: (a) (Cb (X, E), β) is complete. (b) (C(X, E), k) is complete. Proof. (a) Let {fα : α ∈ I} be a β-Cauchy net in Cb (X, E). Since p ≤ β, {fα } is a p-Cauchy net in C(X, E); that is, for each x ∈ X, {fα (x)} is a Cauchy net in E. Since E is complete, for each x ∈ X, f (x) := limα fα (x) β exists in E. We need to show that fα −→ f and that f is continuous and β bounded. Now, fα −→ f , as follows. Let ϕ ∈ Bo (X), 0 ≤ ϕ ≤ 1, and let W ∈ W be closed. Since {fα } is β-Cauchy, there exists αo ∈ I such that ϕ(x)[fα (x) − fγ (x)] ∈ W for all α, γ ≥ αo and x ∈ X. Fix any α ≥ αo . Since W is closed and, for each x ∈ X, fγ (x) −→ f (x), we have ϕ(x)[fα (x) − f (x)] ∈ W for x ∈ X. Thus ϕ(x)[fα (x) − f (x)] ∈ W for all α ≥ αo and all x ∈ X; i.e. fα − f ∈ N(ϕ, W ) for all α ≥ αo . β

This proves that fα −→ f . Next, f is continuous as follows. Since X is a k-space, it suffices to show that, for any compact K ⊆ X, f |K is continuous. Let xo ∈ K and W ∈ W. We show that there exists a neighborhood G of xo in X such that f (y) ∈ f (xo ) + W for all y ∈ G ∩ K.

34

2. COMPLETENESS IN FUNCTION SPACES β

[Choose a balanced V ∈ W such that V + V + V ⊆ W . Since fα −→ f k and k ≤ β, fα −→ f and so there exists an αo ∈ I such that fα (x) − f (x) ∈ V for all α ≥ αo and all x ∈ K.

(1)

Since fαo is continuous at xo , there exists a neighborhood G of xo in X such that (2) fαo (y) ∈ fαo (xo ) + V for all y ∈ G. Then, for any y ∈ G ∩ K, using (1) and (2), f (y) − f (xo ) = [f (y) − fαo (y)] + [fαo (y) − fαo (xo )] + [fαo (xo ) − f (xo )] ∈ V + V + V ⊆ W.] Finally, f is bounded as follows. Suppose f is unbounded. Then there exists a W ∈ W such that f (X) is not contained in nW for all n ≥ 1. Hence, for each n ≥ 1, there exists xn ∈ X such that f (xn ) ∈ / nW . Choose a balanced V ∈ W such that V + V ⊆ W . Let ∞ X 1 χ{xn } . ψ= n n=1 β

Then ψ ∈ Bo (X), and so, since fα −→ f , there exists an αo ∈ I such that ψ(x)[fα (x) − f (x)] ∈ V for all α ≥ αo and x ∈ X. In particular, since ψ(xn ) = n1 , fαo (xn ) − f (xo ) ∈ nV for all n ≥ 1. Since fαo is bounded, there exists an integer m > 1 such that fαo (X) ⊆ mV . Then f (xm ) = [f (xm ) − fαo (xm )] + fαo (xm ) ∈ mV + mV ⊆ mW, a contradiction. Hence f is bounded. (b) Let {fα } be a k-Cauchy net in C(X, E), and let f (x) := lim fα (x), x ∈ X. α

Using the arguments of part (a) with replacing ϕ by χK , K a compact k set in X, it easily follows that fα −→ f and f is continuous. Hence C(X, E) is k-complete. Theorem 2.2.2. [KR91] Let X be a k-space and E a complete TVS. Then (a) (Cb (X, E), σ) is complete. (b) If k ≤ σo , then (Cb (X, E), σo ) is complete.


35

Proof. (a) Let {fα } be a σ-Cauchy net in (Cb (X, E)) and, for each x ∈ X, let f (x) := lim fα (x), x ∈ X. α

As in the proof of Theorem 2.2.1, since X is a k-space and k 6 σ, it is σ easily seen that fα −→ f and that f is continuous on X. Suppose f is not bounded. Then there exist a W ∈ W and a sequence {xn } in X such that f {xn } ∈ / nW for all n ≥ 1. Let A = {xn } and choose a V ∈ W such that V + V ⊆ W . There exists an index αo such that fα (x) − fα (x) ∈ V for all α ≥ αo and x ∈ A. Let λ be a positive number such that fα (X) ⊆ λV . Then, for n > λ, f (xn ) = [f (xn ) − fαo (xn )] + fαo (xn ) ∈ V + λV ⊆ nW, a contradiction. Hence (Cb (X, E), σ) is complete. (b) The proof is almost identical to the one given for (a).

Theorem 2.2.3. [KR91] Let E be a sequentially complete TVS. Then (Cb (X, E), σ) and (Cb (X, E), σo ) are sequentially complete. Proof. Let {fn } be a σ-Cauchy sequence in Cb (X, E), and let f (x) := lim fα (x), x ∈ X. α

σ

As in the proof of Theorem 2.2.2, we see that fn −→ f and that f is bounded. Suppose that f is not continuous at some xo ∈ X. Then, without loss of generality, we may assume that there exist a net {xλ : λ ∈ I} in X, with xλ −→ xo , and a W ∈ W such that f (xλ ) − f (xo ) ∈ / W for all λ ∈ I.

(1)

Choose a V ∈ W such that V + V + V ⊆ W . The functions fn (n = 1, 2, ...) are continuous and so, for each positive integer n, there exists an index λn such that fn (xλ ) − fn (xo ) ∈ V for all λ ≥ λn . Let A = {xo , xλ1 , xλ2 , ...}, a σ-compact set. Choose an integer m such that fn − f ∈ N(A, V ) for all n ≥ m. Then, for n ≥ m, f (xλn ) − f (xo ) = [f (xλn ) − fn (xλn )] + [fn (xλn ) − fn (xo )] +[fn (xo ) − fn (xo )]

36


∈ V + V + V ⊆ W, a contradiction to (1). Hence (Cb (X, E), σ) is sequentially complete. The sequential completeness of (Cb (X, E), σo ) may be proved in the same way. Theorem 2.2.4. [KR91] Let E be a complete locally bounded TVS. If (Cb (X, E), σ) (resp. (Cb (X, E), σo )) is quasi-complete, then it is complete. Proof. Suppose that (Cb (X, E), σ) is quasi-complete, and let {fα } be a σ-Cauchy net in Cb (X, E). Then, as in Theorem 2.1.3, there exists a σ bounded E-valued function f on X such that fα −→ f . We show that f is continuous, as follows. Let V ∈ W be bounded. There exists a closed shrinkable neighborhood S of 0 in E such that S ⊆ V ; the Minkowski functional ρS is continuous (§ A.4). Since V is bounded, we may choose a real number λ ≥ 1, such that V + V ⊆ λS, and, since f is bounded, we may assume that r f (X) ⊆ S for some r ≥ 1. λ We define a function hr : E −→ E by x if x ∈ rS, hr (x) = r x if x ∈ E\rS. ρS (x)

Since ρS is continuous and ρS (x) = r if x ∈ ∂(rS), it follows from Map Gluing Theorem (§ A.1) that hr is continuous. Now hr (E) ⊆ rS and so the net {hr ◦ fα } is u-bounded. Furthermore, {hr ◦ fα } is σ-Cauchy, as follows. Let A be σ-compact and let W ∈ W be any neighborhood of 0 in E. Choose t ≥ 1 such that V + V ⊆ tW . There exists an index αo such that 1 fα − f ∈ N(A, V ) for all α ≥ αo . tλ This implies that fα (A) ⊆ rS for α ≥ αo . Hence, for any x ∈ A and α, γ ≥ αo , 1 (V + V ) ⊆ W. tλ Since (Cb (X, E), σ) is quasi-complete there exists a function g in Cb (X, E) such that σ hr ◦ fα −→ g. hr (fα (x)) − hγ (fr (x)) = fα (x) − fα (x) ∈

σ

The above argument also shows that hr ◦ fα −→ f , and so f = g. Thus f is continuous.


37

Assuming quasi-completeness of (Cb (X, E), σo ), its completeness may be proved by using a similar argument as above.

38


3. Completion of (Cb (X, E), β) We now consider the completion of (Cb (X, E), β), where X is not necessarily a k-space and E is not necessarily complete. Following Katsaras [Kat81], let Bck (X, E) = {f ∈ B(X, E) : f |K is continuous for each compact K ⊆ X} On Bck (X, E) we consider the linear topology βc which has as a base of neighborhoods of 0 consisting of all sets of the form {f ∈ Bck (X, E) : (ϕf )(X) ⊆ W }, where W is a neighborhood of 0 in E and ϕ ∈ Bo (X). Clearly the topology induced on Cb (X, E) by βc is the strict topology β. Theorem 2.3.1. [Kat81] If E is complete, then (Bck (X, E), βc ) is also complete. Proof. Let (fα ) be a βc -Cauchy net in Bck (X, E). Since p ≤ βc and E is complete, there exists an E-valued function f on X such that f (x) = limα fα (x) for each x ∈ X. Now, by the argument similar to the one used in the proof of Theorem 2.2.1, it is easy to verify that f ∈ Bck (X, E) and βc -limα fα = f. Theorem 2.3.2. [Kat81] If (Cb (X, E), β) is complete, then E is complete. Proof. Let {sa } be a Cauchy net in E. For each α, let fa : X → E be the constant function given by fa (x) = sa , x ∈ X. Clearly, the net {fa } is Cauchy in (Cb (X, E), β). Since (Cb (X, E), β) is complete, there exists an f ∈ Cb (X, E) such that fa → f in the topology β. Hence, for any x ∈ X, limα sa = limα fa (x) = f (x). ∧

Definition. Let E be the unique (up to a topological isomorphism) Hausdorff completion of E. We may consider E as a dense subspace of ∧

∧

∧

E. Let βc denote the linear topology on Bck (X, E) which has as a base of neighborhoods of 0 consisting of all sets of the form ∧

∧

U(ϕ, W ) = {f ∈ Bck (X, E) : (ϕf )(X) ⊆ W }, ∧

where ϕ ∈ Bo (X) and W is a neighborhood of 0 in E. Using Theorem 2.3.1, we have the following. Theorem 2.3.3. [Kat81] With the above notations, the completion ∧

∧

of (Cb (X, E), β) is the βc -closure of Cb (X, E) in Bck (X, E) with the induced topology.

3. COMPLETION OF (Cb (X, E), β)

39

∧

∧

Theorem 2.3.4. Let Bck (X, E) and βc be as above and suppose ∧

that, for every compact subset K of X, the space C(K) ⊗ E is u-dense ∧

∧

∧

in C(K, E). Then (Bck (X, E), βc ) is the completion of (Cb (X, E), β). ∧

∧

Proof. Let f ∈ Bck (X, E). We need to show that f belongs to the ∧

∧

βc -closure of Cb (X, E). Consider any βc -neighborhood U(ϕ, W ) of 0 in ∧

Bck (X, E), where ϕ ∈ Bo (X), 0 ≤ ϕ ≤ 1, and W is a neighborhood of ∧

∧

0 in E. Choose an open balanced neighborhood V of 0 in E such that V + V + V + V ⊆ W. Let λ > 1 be such that f (X) ⊆ λV . There exists a compact subset K of X such that 1 ϕ(x) < for all x ∈ X \ K. λ ∧

∧

Since f |K ∈ C(K, E), there exists h ∈ C(K) ⊗ E such that (h−f )(K) ⊆ 1 V . We have λ 1 h(K) ⊆ f (K) + V ⊆ λV + V ⊆ λV + λV. λ ∧

There exists an extension h1 ∈ Cb (X) ⊗ E, h1 = h on K, h1 (X) ⊆ ∧

λV + λV . Since E is dense in E, there exists g ∈ Cb (X) ⊗ E such that (g − h1 )(X) ⊆ V . Now (h1 − f )(X) ⊆ h1 (X) − f (X) ⊆ λV + λV + λV. Let x ∈ X. If x ∈ K, ϕ(x)[g(x) − f (x)] = ϕ(x)[g(x) − h1 (x)] + ϕ(x)[h1 (x) − f (x)] ∈ ϕ(x)V + V ⊆ V + V ⊆ W. If x ∈ X\K, ϕ(x)[g(x) − f (x)] = ϕ(x)[g(x) − h1 (x)] + ϕ(x)[h1 (x) − f (x)] 1 ∈ ϕ(x)V + [λV + λV + λV ] ⊆ W. λ ∧

∧

∧

Therefore g −f ∈ U (ϕ, W ). Thus, by the above theorem, (Bck (X, E), βc ) is the completion of (Cb (X, E), β). ∧

∧

Corollary 2.2.5. [Kat81] (Bck (X, E), βc ) coincides with the completion of (Cb (X, E), β) in each of the following cases: (1) X has finite covering dimension.

40


(2) Every compact subset of X has finite covering dimension. (3) E is a complete metrizable TVS with a basis. (4) E is locally convex. (5) E is a complete TVS and has the approximation property. Proof. The proof follows from the above theorem and a later result, corollary of the Stone-Weierstrass theorem (section 4.1), by which each of the conditions (1)-(5) implies that, for every compact subset K of X, ∧

∧

the space C(K) ⊗ E is u-dense in C(K, E).

4. COMPLETENESS IN THE WEIGHTED TOPOLOGY

41

4. Completeness in the Weighted Topology We now consider completeness in the weighted topology. Let V be a Nachbin family on X, and let F Vo (X, E) denote the set of all E-valued functions f on X such that, for each v ∈ V , vf is bounded and vanishes at infinity. We may define the weighted topology ωV on F Vo(X, E) in the same way as in Section 1.2: Definition. ωV is the linear topology F Vo (X, E) which has a base of neighborhoods of 0 consisting of all sets of the form N(v, G) = {f ∈ F Vo(X, E) : (vf )(X) ⊆ G}, where v ∈ V and G ∈ W. Clearly, (F Vo (X, E), ωV ) is Hausdorff if V > 0. Theorem 2.4.1. [Pro71b, Kh85b] Let V be a Nachbin family on X with V > 0, and suppose E is complete. Then (F Vo (X, E), ωV ) is complete; hence (CVo (X, E), ωV ) is complete iff it is closed in (F Vo (X, E), ωV ). Proof. Let {fα } be an ωV -Cauchy net in F Vo (X, E). Since V > 0, ρ ≤ ωV and so {fα } is ρ-Cauchy. Since E is complete, for each x ∈ X, ωV f . Let v ∈ V , f (x) = limα fα (x) exists, and it is easily shown that fα −→ G ∈ W. Choose a balanced H ∈ W with H + H ⊆ G. There exists an index αo such that v(x)(fα (x) − f (x)) ∈ H for all α ≥ αo and x ∈ X. Choose r ≥ 1 with (vfα )(X) ⊆ rH. Then, for any x ∈ X, v(x)f (x) = v(x)(f (x) − fαo (x)) + v(x)fαo (x) ∈ H + rG ⊆ rG, and so vf is bounded. Next, with v, G, H, and αo as above, choose a / K. Then, for compact set K ⊆ X such that v(x)fαo (x) ∈ H for all x ∈ x∈ / K, we have v(x)f (x) ∈ G. Thus f ∈ F Vo (X, E). Theorem 2.4.2. [Pro71b, Kh85b] Suppose E is complete, and U and V are Nachbin families on X with U ≤ V and V > 0. (a) If CUo (X, E) is closed in (F Uo (X, E), ωU ), then (CVo (X, E), ωV ) is complete. (b) If (F Uo (X, E), ωU ) is complete (quasi-complete), so is (CUo (X, E), ωV ). Proof. (a) In view of Theorem 2.4.1, we need to show that CVo (X, E) is ωV -closed in F Vo (X, E). Let f belong to the ωV -closure of CVo (X, E) ωV f. Since in F Vo (X, E). There exists a net {fα } ⊆ CVo (X, E) with fα −→ ωU f. Since CVo (X, E) ⊆ CUo (X, E) and U ≤ V , ωU ≤ ωV and so fα −→

42


CUo (X, E) is ωV -closed in F Uo (X, E), we have f ∈ F Uo (X, E). Hence f is continuous and so f ∈ CVo (X, E). (b) If (CUo (X, E), ωU ) is complete, then it follows immediately from (a) that (CVo (X, E), ωV ) is complete. Suppose now that (CUo (X, E), ωV ) is quasi-complete, and let {fα } be a bounded Cauchy net in (CUo (X, E), ωV ). ω

V By Theorem 2.4.1, there exists an f ∈ (F Uo (X, E) such that fα −→ f. ωU Since U ≤ V, {fα } ⊆ F Vo (X, E) such that fα −→ f . Clearly, {fα } is ωU bounded. Then, by hypothesis, f ∈ CUo (X, E). Hence f ∈ CVo (X, E).

Corollary 2.4.3. [Pro71b, Kh85b] If X is a k-space and E is complete, then (Cb (X, E), β) is complete. Proof. Let V = So+ (X). Then ωV = β on CVo (X, E) = Cb (X, E). Further, for each x ∈ X, χ{x} ∈ V , and so V > 0. Take U = Kc+ (X). Then (CUo (X, E), ωU ) = (C(X, E), k) which is complete since X is a k-space (see Theorem 2.2.1). So, by Theorem 2.4.2, (Cb (X, E), β) is complete.


43

5. Notes and Comments Section 2.1. We present some classical results on completeness of the spaces (B(X, E), u), (Cb (X, E), u) and (Co (X, E), u), as given in [KN63] Section 2.2. Various completeness properties (such as completeness, sequential completeness and quasi-completeness) of function spaces with the topologies u, β, σ and k are considered. These are taken from [Kh79, KR91] which generalize earlier results given in [Buc58, GulD72]. Section 2.3. This includes some results, due to Katsaras [Kat81], on the completion of (Cb (X, E), β), where X is not necessarily a k-space and E is not necessarily complete. Section 2.4. Completeness in the weighted spaces CVb (X, E) and CVo (X, E) is considered. These were obtained by Summers [SumW69] for E a scalar field, by Prolla [Pro71b] for E a locally convex TVS and by Khan [Kh85] for E any TVS.

CHAPTER 3

The Arzela-Ascoli type Theorems In this chapter we study Arzela-Ascoli type theorems for the compactopen, strict and weighted topologies. The abstract version of the ArzelaAscoli theorem was first given by Myers [My46] which was later improved by Gale [Gal50]. The name k-space also first appeared in the paper [Gal50]. We include here some related results from [KN63, Kh79, RS88, KO05].

45

46

3. THE ARZELA-ASCOLI TYPE THEOREMS

1. Arzela-Ascoli Theorem for k and β Topologies In this section, we present Arzela-Ascoli type theorems for the compactopen and strict topologies. Definition. Let X be a topological space and E a TVS with a base W of neighborhoods of 0. A subset A of C(X, E) is said to be equicontinuous at xo ∈ X if, for each W ∈ W, there exists a neighborhood G(xo ) of xo such that f (x) − f (xo ) ∈ W for all x ∈ G(xo ) and f ∈ A. A is said to be equicontinuous on X if it is equicontinuous at each point of X. The following version of the Arzela-Ascoli type theorem for the compactopen topology is given in [KN63, p. 81]. Theorem 3.1.1. [KN63, p. 81] Let X be a Hausdorff k-space and E a Hausdorff TVS , and let A ⊆ C(X, E). Then A is k-compact iff the following conditions are satisfied. (i) A is k-closed; (ii) A(x) = {f (x) : f ∈ A} is relatively compact in E for each x ∈ X; (iii) A is equicontinuous on each compact subset of X. For the readers’ benefit and interest, we present here a well-known version of above theorem in the more general setting of uniform spaces. Its proof is adapted from [Eng88, p. 440-443; Will70, § 43]. Recall that, if Y is a non-empty set, then, for any A, B ⊆ Y × Y, we define A−1 = {(y, x) : (x, y) ∈ A}, A ◦ B = {(x, y) ∈ Y × Y : ∃ z ∈ Y such that (x, z) ∈ A and (z, y) ∈ B}. If A = B, we shall write A ◦ A = A2 . Note that, (x, z) ∈ A, (z, y) ∈ B ⇒ (x, y) ∈ A ◦ B. A subset H of subsets of Y × Y is called a uniformity (or a uniform structure) on Y if (u1 ) △(Y ) = {(x, x) : x ∈ Y } ⊆ A for all A ∈ H. (u2 ) A ∈ H =⇒ A−1 ∈ H, i.e. each A ∈ H is symmetric. (u3 ) If A ∈ H, ∃ some B ∈ H such that B 2 ⊆ A. (u4 ) If A, B ∈ H, ∃ some C ∈ H such that. C ⊆ A ∩ B. (u5 ) If A ∈ H and A ⊆ B, then B ∈ H.

1. ARZELA-ASCOLI THEOREM FOR k AND β TOPOLOGIES

47

In this case, the pair (Y, H) is called a uniform space. The members of H are called vicinities of H. For any x ∈ Y and A ∈ H, let NA (x) = {y ∈ Y : (x, y) ∈ A}. Then, for each x ∈ Y , the collection {NA (x) : A ∈ H} satisfies all the conditions of a neighborhood system at x. Consequently, there exists a unique topology τH on Y such that, for each x ∈ Y , the collection {NA (x) : A ∈ H} forms a local base at x. In this case, the topology τH on Y is said to be the topology induced by (or derived from) the uniformity H. For any net {yα } ⊆ (Y, H), we say that yα → y ∈ Y if, for each V ∈ H, there exists an αo ∈ I such that (yα , y) ∈ V for all α ≥ αo . Every TVS (E, τ ) is a uniform space. [In fact, let W be a base of neighborhoods of 0 in E. For each V ∈ W, let AV = {(x, y) ∈ E × E : x − y ∈ V }. Then H = {AV : V ∈ W} is a uniformity on E. ] Definition. Let X be a Hausdorff space and (Y, H) a uniform space. and let S = S(X) be a certain collection of subsets of X. For any K ∈ S(X) and D ∈ H, let M(K, D) = {(f, g) ∈ C(X, Y )×C(X, Y ) : (f (x), g(x)) ∈ D for all x ∈ K}. Then the collection {M(K, D) : K ∈ S(X) and D ∈ H} form a subbase for a uniformity, called the uniformity of uniform convergence on the sets in S(X) induced by H. The resultant topology τS on C(X, Y ) is called the topology of uniform convergence on the sets in S(X). (1). If S(X) = {X}, then the τS -topology on C(X, Y ) is called the uniform topology and is denoted by u. (2). If S(X) is the collection of all compact subsets of X, then the τS topology on C(X, Y ) is called the compact-open topology and is denoted by k. (3). If S(X) is the collection of all finite (or singleton) subsets of X, then the τS -topology on C(X, Y ) is called the pointwise topology and is denoted by p Clearly p ≤ k ≤ u on C(X, Y ). and k = u if X is compact. Alternate Definition. [Edw65, p. 33] Let X be a Hausdorff space and (Y, H) a uniform space having a base B of vicinities of H, and let

48


S = S(X) be a collection of subsets of X. K ∈ S(X) and A ∈ B (or A ∈ H), let

For any f ∈ C(X, Y ),

N(f, K, A) = {g ∈ C(X, Y ) : (g(x), f (x)) ∈ A for all x ∈ K}. Then the collection {N(f, K, A) : f ∈ C(X, Y ), K ∈ S(X), A ∈ B} form a subbase for a topology τS on C(X, Y ), called the topology of uniform convergence on the sets in S(X). Definition. [Eng88, p. 442] Let X be a Hausdorff space and (Y, H) a uniform space. A subset A of C(X, Y ) is said to be equicontinuous at xo ∈ X if, for each D ∈ H, there exists a neighborhood G(xo ) of xo such that (f (x), f (xo )) ∈ D for all x ∈ G(xo ) and f ∈ A. A is said to be equicontinuous on X if it is equicontinuous at each point of X. Lemma 3.1.2. Let X be a Hausdorff space and (Y, H) a uniform p space, and let A ⊆ C(X, Y ). If A is equicontinuous on X, then A is also equicontinuous on X. Proof. Let xo ∈ X and D ∈ H. We show that there exists a neighborhood G(xo ) of xo such that p

(g(y), g(xo)) ∈ D for all y ∈ G(xo ) and g ∈ A .

(1)

Choose a symmetric V ∈ H with V ◦V ◦V ⊆ D. Since A is equicontinuous on X, there exists a neighborhood G(xo ) of xo such that (f (y), f (xo)) ∈ V for all y ∈ G(xo ) and f ∈ A. p

(2) p

Let f ∈ A , and let {fα : α ∈ I} be a net in A such that fα → f. Fix any y ∈ G(xo ). Choose α0 ∈ I such that (fα0 (y), f (y)) ∈ V and (fα0 (xo ), f (xo )) ∈ V.

(3)

Now, by (2) and (3), (f (y), fα0 (y)), (fα0 (y), fα0 (xo )), (fα0 (xo ), f (xo )) ∈ V, and so (f (y), f (xo)) ∈ V ◦ V ◦ V ⊆ D. p This proves (1), Thus A is equicontinuous on X.

Lemma 3.1.3 Let X be a Hausdorff space and (Y, H) a uniform space, and let A ⊆ C(X, Y ). If A is equicontinuous on each compact subset of X, then: (a) The topologies k and p coincide on A. k p (b) A = A .


49

Proof. The proofs of (a) and (b) are similar and so we shall only give the proof of (b). k p p k Since p ⊆ k, A ⊆ A and so we need to prove that A ⊆ A . Let p f ∈ A and let {fα : α ∈ I} be a net in A such that p-limfα = f. α Let K ⊆ X be compact and D ∈ H. Choose a symmetric V ∈ H with V ◦ V ◦ V ⊆ D. Since p-limfα = f , for each x ∈ K, there exists αx ∈ I α such that (fα (x), f (x)) ∈ V for all α ≥ αx .

(1) p

Since A is equicontinuous subset on K, by Lemma 3.1.2, A is also equicontinuous on K. Hence there exists a neighborhood G(x) of x such that p

(g(y), g(x)) ∈ V for all y ∈ G(x) ∩ K and g ∈ A .

(2)

Choose a finite subcover {G(xi ) : i = 1, ..., m} of {G(x) : x ∈ K} for K. Choose α0 ∈ I such that α0 ≥ αxi , i = 1, ..., m. Let y ∈ K. Then y ∈ G(xq ) ∩ K for some q (1 ≤ q ≤ m). Hence, for all α ≥ α0 , it follows from (1) and (2) that (fα (y), fα(xq )), (fα (xq ), f (xq )), (f (xq ), f (y)) ∈ V ; consequently, (fα (y), f (y)) ∈ V ◦ V ◦ V ⊆ D. k

This implies that (fα , f ) ∈ M(K, D) for all α ≥ α0 . Hence f ∈ A . We now establish a version of the Arzela-Ascoli theorem for (C(X, Y ), k). Theorem 3.1.4. Let X be a Hausdorff k-space and (Y, H) a Hausdorff uniform space, and let A ⊆ C(X, Y ). Then A is k-compact iff the following conditions are satisfied. (i ) A is k-closed; (ii ) A(x) = {f (x) : f ∈ A} is relatively compact in Y for each x ∈ X; (iii ) A is equicontinuous on each compact subset of X. Proof. (⇒) Suppose A is k-compact in C(X, Y ). Then conditions (i) - (iii) hold, as follows. (i) Since Y is Hausdorff, C(X, Y ), k) is also Hausdorff. So A, being a compact subset of the Hausdorff space, is k-closed. (ii) Let x be any point in X. Consider the evaluation map δx : C(X, Y ) → Y given by δx (f ) = f (x), f ∈ C(X, Y ).

50


Now δx is p-continuous, as follows. For any f ∈ C(X, Y ) and D ∈ H. Let N(f, {x}, D) = {g ∈ C(X, Y ) : (g(x), f (x)) ∈ D}. It is clear that N(f, {x}, D) is a pneighborhood of f in C(X, Y ) and δx (N(f, {x}, D)) = {g(x) : (g(x), f (x)) ∈ D} ⊆ ND (f (x)). Hence δx is p-continuous. Since p ≤ k and A is k-compact, it follows that A is p-compact. Thus δx (A) = A(x) is relatively compact in Y . (iii) Suppose K is any compact subset of X. let xo ∈ K and D ∈ H. Choose a symmetric V ∈ H with V ◦ V ◦ V ⊆ D. Now, for each f ∈ A, there exists a neighborhood G1 (xo , f ) of xo such that (f (x), f (xo )) ∈ D for all x ∈ G1 (xo , f ). Since K, being compact and Hausdorff, is regular, there exists a closed neighborhood G(xo , f ) of xo in K such that G(xo , f ) ⊆ G1 (xo , f ) ∩ K. Let N(f, G(xo , f ), V ) = {g ∈ C(X, Y ) : (g(x), f (x)) ∈ V for all x ∈ G(xo , f )}. a k-open neighborhood of f. Since A is k-compact, its open cover {N(f, G(xo , f ), V ) : f ∈ A} has a finite subcover {N(fi , G(xo , fi ), V ), i = 1, ..., m}. Let f ∈ A, so that f ∈ N(fj , G(xo , fj ) for some j (1 ≤ j ≤ m). Then, for any x ∈ ∩m i=1 G(xo , fi ), (f (x), fj (x)), (fj (x), fj (xo )), (fj (xo ), f (xo )) ∈ V. and so (f (x), f (xo )) ∈ V ◦ V ◦ V ⊆ D. So A is equicontinuous at xo and hence on K. [⇐] Suppose that A satisfies conditions (i) - (iii). Since A is equicontinuous on each compact subset of X, it follows from Lemmas 3.1.2 and p p p k 3.1.3 that A is equicontinuous, k = p on A and A = A . Since A is k-closed, to show that A is k-compact, it suffices to show that A is p-compact.Q We may regard C(X, Y ) as being embedded in the product space x∈X Yx where each Yx =Q Y via the correspondence f → (f (x))x∈X . The product topology on x∈X Yx induces on C(X, Y ) the topology p. In particular, we may regard AQ as being embedded in Q A(x) is compact. x∈X A(x). By (ii) and the Tychonoff theorem, Q x∈X Hence it suffices to show that A is p-closed in x∈X A(x). Q Let g belong to the p-closure of A in x∈X A(x). Since X is a kspace, to show that g is continuous on X, it is sufficient to show that g


51

is continuous on any compact subset K of X. Let xo ∈ K and D ∈ H. Choose a symmetric V ∈ H with V ◦V ◦V ⊆ W. Since A is equicontinuous on X, there exists a neighborhood G(xo ) of xo such that (f (y), f (xo)) ∈ V for all y ∈ G(xo ) ∩ K and f ∈ A. Q Since g belongs to the p-closure of A in x∈X A(x), for each x ∈ X, there exists fx ∈ A such that (fx (x), g(x)) ∈ V.

Then, for any y ∈ G(xo ) ∩ K, (g(y), fy (y)), (fy (y), fy (xo )), (fy (xo ), g(xo )) ∈ V. and so (g(y), g(xo)) ∈ V ◦ V ◦ V ⊆ D. p

p

k

Therefore g ∈ C(X, Y ). Hence g ∈ A . But A = A = A (since A is Q k-closed). So g ∈ A, showing that A is p-closed in x∈X A(x).

Note. In the case of Y = E, a TVS, the above theorem reduces to Theorem 3.1.1. We now present an analogue of the above theorem for the β-topology on Cb (X, E) with E a TVS. Here we use the terminology of Chapter 1. Theorem 3.1.5. [Kh79] Let X be a Hausdorff k-space and E a Hausdorff TVS. Then a subset A of Cb (X, E) is β-compact iff the following conditions hold: (i) A is β-closed and β-bounded; (ii) A(x) is relatively compact in E for each x ∈ X; (iii) A is equicontinuous on each compact subset of X. Proof. Suppose A is β-compact in Cb (X, E). Then conditions (i) (iii) hold, as follows. (i) Since E is Hausdorff, Cb (X, E), β) is also Hausdorff. So A, being a compact subset of the Hausdorff space, is β-closed. Further, A is bounded since a compact set in a TVS is always bounded. (ii) Let x be any point in X. Consider the map δx : Cb (X, E) → E given by δx (f ) = f (x), f ∈ Cb (X, E). Now δx is p-continuous, as follows. For any f ∈ Cb (X, E) and W ∈ W. Let N(f, {x}, W ) = {g ∈ Cb (X, E) : g(x) − f (x) ∈ W }. It is clear that N(f, {x}, W ) is a pneighborhood of f in Cb (X, E) and δx (N(f, {x}, W )) ⊆ f (x) + W.

52


Hence δx is p-continuous. Since p ≤ β and A is β-compact, it follows that A is p-compact. Thus δx (A) = A(x) is relatively compact in E. (iii) Since k ≤ β and A is β-compact, it follows by Theorem 3.1.1 that A is equicontinuous on each compact subset of X. Conversely, suppose that a subset A of Cb (X, E) satisfies conditions (i) - (iii). Since A, being β-bounded, is u-bounded, the topologies β and k coincide on A (Theorem 1.1.8, (c), (d)). Thus, to show that A is β-compact, it is only necessary to show that A is k-compact. Now, by using the same argument as the one use to prove Theorem 1.1.8 (c), we can show that the β and k closures of A are the same. Consequently, A is k-closed. In view of Theorem 3.1.1, this fact together with conditions (ii) and (iii) imply that A is k-compact.

2. ARZELA-ASCOLI THEOREM FOR WEIGHTED TOPOLOGY

53

2. Arzela-Ascoli Theorem for Weighted Topology In this section, we present the Arzéla-Ascoli type theorems which characterize precompact subsets of weighted functions spaces CVo (X, E) and CVpc (X, E) in the non-locally convex setting. When the range space E happens to be quasi-complete, these characterizations become criteria for relative compactness. As in section 1.2, let X be a completely regular Hausdorff space, E is non-trivial Hausdorff TVS with a base W of closed balanced neighborhoods of 0 and V a Nachbin family of non-negative upper semicontinuous function on X. We also assume that V > 0; i.e., for each x ∈ X, there exists a v ∈ V with v(x) > 0. Further, let CVpc (X, E) = {f ∈ C(X, E) : (vf )(X) is precompact in E for all v ∈ V }. Clearly, CVo (X, E) ⊆ CVpc (X, E) ⊆ CVb (X, E). The weighted topology ωV on CVb (X, E) has a base of neighborhoods of 0 consisting of all sets of the form N(v, G) = {f ∈ CVb (X, E) : (vf )(X) ⊆ G}, where v ∈ V and G ∈ W. Note that: (1) If V = K + (X), then CVpc (X, E) = Cb (X, E) and ωV is the uniform topology u. (2) If V = So+ (X), then CVpc (X, E) = Cb (X, E) and ωV is the strict topology βo . (3) If V = Kc+ (X), then CVpc (X, E) = CVb (X, E) = CVo (X, E) = C(X, E) and ωV is the compact-open topology k. (4) If V = Kf+ (X),then CVpc (X, E) = CVb (X, E) = CVo (X, E) = C(X, E) and ωV is the pointwise topology p. The assumption V > 0 implies that p ≤ ωV . Recall that p ≤ k on C(X, E) and k ≤ βo ≤ u on Cb (X, E). Definition. Let E and F be TVSs. For any collection G of subsets of E, CLG (E, F ) denotes the subspace of CL(E, F ) consisting of those T which are bounded on the members of G together with the topology tG of uniform convergence on the elements of G. This topology has a base of neighborhoods of 0 consisting of all sets of the form U(D, W ) := {T ∈ CL(E, F ) : T (D) ⊆ W }, where D ∈ G and W is a neighborhood of 0 in F (see § A.3).

54


Definition. If V is a Nachbin family on X, a net {xα : α ∈ I} is called a V -net if it is contained in Sv,1 := {x ∈ X : v(x) ≥ 1} for some v ∈ V . We hence get the classical VR -spaces introduced by Bierstedt [Bie75b]: X is said to be a VR -space if a function f : X → R is continuous whenever, for each v ∈ V, the restriction of f to Sv,1 is continuous. If V = K + (X), then X is a VR -space means X is a kR -space. Definition. For any x ∈ X, let δx : CVb (X, E) → E denote the evaluation map δx (f ) = f (x), f ∈ CVb (X, E). Clearly, δx ∈ CL(CVb (X, E), E). Next, we define ∆ : X → CL(CVb (X, E), E) as the evaluation map given by ∆(x) = δx , x ∈ X. Let the subscript pc in CLpc (CVb (X, E), E) stand for the topology of uniform convergence on precompact subsets of CVb (X, E). Lemma 3.2.1. [RS88, KO05] The evaluation map ∆ : X → CLpc (CVb (X, E), E) is continuous iff every precompact subset of CVb (X, E) is equicontinuous. Proof. (⇒) Suppose ∆ : X → CLpc (CVb (X, E), E) is continuous, and let A be a precompact subset of CVb (X, E). To show that A is equicontinuous, let xo ∈ X and W ∈ W. Since ∆ is continuous at xo , there exists an open neighborhood G(xo ) of xo in X such that ∆(G(xo )) ⊆ ∆(xo ) + U(A, W ); that is, δx (f ) − δxo (f ) ∈ W for all x ∈ G(xo ) and f ∈ A. Hence f (G) ⊆ f (xo ) + W for all f ∈ A, and so A is equicontinuous. (⇐) Suppose that every precompact subset of CVb (X, E) is equicontinuous. To show that ∆ : X → CLpc (CVb (X, E), E) is continuous, let xo ∈ X and let A be a precompact subset of CVb (X, E) and W a balanced set in W. Since A, being precompact, is equicontinuous (by hypothesis), there exists an open neighborhood G(xo ) of xo in X such that f (G) ⊆ f (xo ) + W for all f ∈ A; that is δx − δxo ∈ U(A, W ) for all x ∈ G. Hence ∆(G(xo )) ⊆ ∆(xo ) + U(A, W ). Theorem 3.2.2. [RS88, KO05] Let X be a VR -space. Then every precompact subset of CVb (X, E) is equicontinuous. Proof. In view of Lemma 3.2.1, it suffices to show that the evaluation map ∆ : X → CLpc (CVb (X, E), E) is continuous. Since X is a VR -spaces, we only need to show that ∆ is continuous on each Sv,1 = {x ∈ X :


55

v(x) ≥ 1}, v ∈ V. Let v ∈ V and x ∈ Sv,1 , and let A be a precompact subset of CVb (X, E) and W ∈ W. Choose a balanced H ∈ W with H + H + H ⊆ W . Since A is precompact, there exist h1 , ..., hn ∈ A such that n [ A ⊆ (hi + N(v, H)). i=1

Since each hi is continuous, there exists a neighborhood Gi of x in X such that hi (y) − hi (x) ∈ H for all y ∈ Gi , i = 1, ..., n.

Let G =

n T

Gi . Now, if y ∈ G ∩ Sv,1 and f ∈ A, then f = hi + g for

i=1

some 1 ≤ i ≤ 1 and g ∈ N(v, H). Hence δy (f ) − δx (f ) = hi (y) + g(y) − hi (x) − g(x) 1 1 = hi (y) − hi (x) + v(y)g(y) − v(x)g(x) v(y) v(x) 1 1 H− H ⊆ H + H − H ⊆ W; ∈ H+ v(y) v(x) that is, δy − δx ∈ U(A, W ) for all y ∈ G ∩ Sv,1 . Hence ∆(G ∩ Sv,1 ) ⊆ ∆(x) + U(A, W ). Theorem 3.2.3. [RS88, KO05] Let X be a VR -space. A subset A of CVo (X, E) is precompact iff the following conditions hold. (i) A is equicontinuous. (ii) A(x) = {f (x) : f ∈ A} is precompact in E for each x ∈ X. (iii) vA vanishes at infinity on X for each v ∈ V ; i.e., given v ∈ V and W ∈ W, there exists a compact set K ⊆ X such that v(y)f (y) ∈ W for all f ∈ A and y ∈ X\K. Proof. (⇒) Suppose A is a precompact subset of CVo (X, E). We verify (i)-(iii), as follows. (i) Since X is a VR -space, by Theorem 3.2.2, A is equicontinuous. (ii) Since V > 0, p ≤ ωV and so A is p-precompact. Hence, for each x ∈ X, A(x) is precompact in E. (iii) Let v ∈ V and W ∈ W. Choose a balanced H ∈ W with H + H ⊆ W . Since A is precompact, there exist h1 , ..., hn ∈ A such that A⊆

n [

(hi + N(v, H)).

i=1

(1)

56


Put K =

n S

{y ∈ X : v(y)hi (y) ∈ / H}. Then K is compact (since each

i=1

hi ∈ CVo (X, E)). Now let f ∈ A and y ∈ X\K. By (1), there exists i ∈ {1, ..., n} such that f ∈ hi + N(v, H). Hence v(y)f (y) = v(y)[f (y) − hi (y)] + v(y)hi (y) ∈ H + H ⊆ W. Thus vA vanishes at infinity on X. (⇐) Suppose (i) - (iii) hold. Since CVo (X, E) ⊆ C(X, E), by (i), A is an equicontinuous subset of C(X, E). Further, since, by (ii), A is pprecompact, it follows that A is a precompact subset of (C(X, E), k) (cf. Lemma 3.1.2). To show that A is precompact in CVo (X, E), let v ∈ V and W ∈ W. Choose a balanced H ∈ W such that H + H ⊆ W . By (iii), there exists a compact K ⊆ X such that v(y)f (y) ∈ H for all f ∈ A and y ∈ X\K.

(2)

Since v is upper-semicontinuous, kvkK = sup{v(y) : y ∈ K} < ∞. Since A is precompact in (C(X, E), k), there exist h1 , ..., hn ∈ A such that A⊆

n [

(hi + N(χK ,

i=1

We claim that

A⊆

n [

1 kvkK + 1

H)).

(3)

(hi + N(v, W )).

i=1

Let f ∈ A and y ∈ X. If y ∈ K, by (3), there exists j ∈ {1, ..., n} such that 1 H ⊆ H. v(y)[f (y) − hj (y)] ∈ v(y) kvkK + 1 If y ∈ X\K, then, for any i ∈ {1, ..., n}, (2) gives v(y)[f (y) − hi (y)] = v(y)f (y) − v(y)hi (y) ∈ H − H ⊆ W. This establishes our claim, and so A is precompact in CVo (X, E). Following [RS88], we shall use the following notation: for any A ⊆ CVpc (X, E), v ∈ V and W ∈ W, let Tx (A, v, W ) = {y ∈ X : v(y)f (y) −v(x)f (x) ∈ W for all f ∈ A}, x ∈ X. Theorem 3.2.4. [RS88, KO05] Let X be a VR -space. Then, for any A ⊆ CVpc (X, E), the following are equivalent: (a) A is precompact. (b) (i) A is equicontinuous; (ii) A(x) is precompact in E for each x ∈ X;


57

(iii) given v ∈ V and W ∈ W, there exists a compact set K ⊆ X such that {Tx (A, v, W ) : x ∈ K} covers X. (c) (i) vA(X) = {v(x)f (x) : x ∈ X, f ∈ A} is precompact in E each v ∈V; (ii) given v ∈ V and W ∈ W, there exists a finite set F ⊆ X such that {Tx (A, v, W ) : x ∈ F } covers X. (d) (i) A(x) is precompact in E for each x ∈ X; (ii) given v ∈ V and W ∈ W, there exists a finite set F ⊆ X such that {Tx (A, v, W ) : x ∈ F } covers X. Proof. (a) ⇒ (b) Since A is a precompact subset of CVpc (X, E), just as in the proof of Theorem 3.2.3, (b)(i) follow from Theorem 3.2.2 and (b)(ii) follows from the fact that V > 0. To prove (b)(iii), let v ∈ V and W ∈ W. Choose a balanced H ∈ W such that H + H + H ⊆ W . Since A is precompact, there exist h1 , ..., hn ∈ A such that n [ (1) A ⊆ [hi + N(v, H)]. i=1

Now, consider the function h : X → E n defined by

h(x) = (h1 (x), h2 (x), . . . , hn (x)). This is a continuous function Q such that (vh)(X) is precompact, for it is contained in the product ni=1 (vhi )(X) which is precompact. Hence for the neighborhood H n , there exists a finite subset F of X such that [ (vh)(X) = ((vh)(x) + H n ) . x∈F

This gives

X⊆

[

Tx ({h1 , . . . , hn }, v, H).

(2)

x∈F

We now show that {Tx (A, v, W ) : x ∈ F } covers X. Fix y ∈ X. By (2), y ∈ Tx ({hi }ni=1 , v, H) for some x ∈ F and so v(y)hi (y) − v(x)hi (x) ∈ H for all i = 1, ..., n.

(3)

Given f ∈ A, by (1), there exists i ∈ {1, ..., n} such that f −hi ∈ N(v, H); i. e., v(z)(f (z) − hi (z)) ∈ H for all z ∈ X. (4) So, by (3) and (4), v(y)f (y) − v(x)f (x) = (v(y)f (y) − v(y)hi (y)) +(v(y)hi (y) − v(x)hi (x)) +(v(x)hi (x) − v(x)h(x)

58


∈ H + H − H ⊆ W. Hence y ∈ Tx (A, v, W ); i.e., (b)(iii) holds. (b) ⇒ (c) Suppose (b) holds. We first note that (b)(i) and (b)(ii) together imply (as in the proof of Theorem 3.2.3) that A is a precompact subset of (C(X, E), k). We now verify (c)(i) and (c)(ii). (c)(i). Let v ∈ V and W ∈ W. Choose a balanced H ∈ W such that H + H + H + H ⊆ W. By (b)(iii), there exists a compact K ⊆ X such that {Tx (A, v, H) : x ∈ K} covers X. (1) Since A is precompact in (C(X, E), k), there exist h1 , ..., hn ∈ A such that n [ 1 A ⊆ [hi + N(χK , H)]. (2) kvk + 1 K i=1 Moreover, since each vhi (K) is precompact in E, there exist {xij : 1 ≤ j ⊆ ni } ⊆ K such that vhi (K) ⊆

ni [

[v(xij )h(xij ) + H].

(3)

j=1

Now, fix any y ∈ X and f ∈ A. By (1), y ∈ Tx (A, v, H) for some x ∈ K and so v(y)f (y) − v(x)f (x) ∈ H for all f ∈ A. (1a) By (2), there exists i ∈ {1, ..., n} such that 1 H. (2a) (f − hi )(K) ⊆ kvkK + 1 By (3), there exists j ∈ {1, ..., n} such that v(x)hi (x) − v(xij )hi (xij ) ∈ H.

(3a)

By (1a), (2a), (3a) v(y)f (y) − v(xij )hi (xij ) =[v(y)f (y) − v(x)f (x)] + v(x)[f (x) − hi (x)] + [v(x)hi (x) − v(xij )hi (xij )] 1 ∈H + v(x) H + H ⊆ W. kvkK + 1 i.e. vA(X) ⊆

ni n S S

[v(xij )hi (xij ) + W ] and vA(X) is precompact in E.

i=1 j=1

(c) (ii

(4)


59

). Let v ∈ V and W ∈ W, and suppose these be same as in the proof n S of (c)(i). Also, let hi and xij be same as above. Let F = {xij : j = i=1

1, ..., ni }. Then, for any fixed y ∈ X and f ∈ A, (3a) and (4) give v(y)f (y) − v(xij )f (xij ) = [v(y)f (y) − v(xij )hi (xij )] +[v(xij )hi (xij ) − v(xij )f (xij )] ∈ (H + H + H) + H ⊆ W ;

that is, y ∈ Tx (A, v, W ) with x ∈ F . Hence {Tx (A, v, W ) : x ∈ F } covers X. (c) ⇒ (d) Since V > 0, p ≤ ωV and so, for any v ∈ V, vA(X) is precompact implies that A(x) is precompact in E for each x ∈ X. (d) ⇒ (a) Suppose (d) holds. Fix any v ∈ V and W ∈ W. Choose a balanced H ∈ W with H + H + H ⊆ W . By (d)(ii) there exists a finite set F ⊆ X such that {Tx (A, v, H) : x ∈ F } covers X. By (d)(i), A is p-precompact, and so there exists hi , ..., hn ∈ A such that n [ 1 A ⊆ (hi + N(χF , H). (1) kvk + 1 F i=1 We claim that A ⊆

n S

(hi + N(v, W )). Fix any f ∈ A. By (1), there

i=1

exists i ∈ {1, ..., n} such that (f − hi )(F ) ⊆

1 kvkF + 1

H).

Then, for any y ∈ X, y ∈ Tx (A, v, H) for some x ∈ F and so v(y)f (y) − v(y)hi (y) = [v(y)f (y) − v(x)f (x)] + v(x)[f (x) − hi (x)] +[v(x)hi (x) − v(y)hi (y)] 1 H−H ∈ H + v(x) kvkF + 1 ⊆ H + H + H ⊆ W. This proves our claim; i.e., (a) holds. In the particular cases of V = Kc+ (X), K + (X) and So+ (X), we mention that Theorems 3.2.3 and 3.2.4 give further improvements of Theorems 3.1.3 and 3.1.4 and are stated as follows: Corollary 3.2.5. Let X be a kR -space and E a quasi-complete TVS . A subset A of (C(X, E), k) is relatively compact iff the following conditions hold: (i) A is equicontinuous on each compact subset of X,

60


(ii) A(x) is relatively compact in E for each x ∈ X. Corollary 3.2.6. Let X be a locally compact space and E a quasicomplete TVS . A subset A of (Co (X, E), u) is relatively compact iff the following conditions hold: (i) A is equicontinuous, (ii) A(x) is relatively compact in E for each x ∈ X, (iii) A uniformly vanishes at infinity on X; i.e., for any W ∈ W, there exists a compact set K ⊆ X such that f (y) ∈ W for all f ∈ A and y ∈ X\K. Corollary 3.2.7. Let X be a kR -space and E a quasi-complete TVS . A subset A of (Cb (X, E), βo ) is relatively compact iff the following conditions hold: (i) A is equicontinuous on each compact subset of X, (ii) A(X) is relatively compact in E, (iii) A is uniformly bounded (i.e., A(X) is bounded in E). Corollary 3.2.8. Let E be a quasi-complete TVS . A subset A of (Cpc(X, E), u) is relatively compact iff the following conditions hold: (i) A(X) is relatively compact in E, (ii) given W ∈ W, there exists a finite open cover {Ki : i = 1, ..., n} of X such that, for any i ∈ {1, ..., n} and x, y ∈ Ki , f (x) − f (y) ∈ W for all f ∈ A.


61

3. Notes and Comments Section 3.1. Theorem 3.1.1. is a version of the Arzela-Ascoli Theorem for the k-topology, given in the book of Kelley and Namioka [KN63, p. 81] without a proof. For the readers’ benefit and interest, we present a well-known version of this theorem (Theorem 3.1.4) in the more general setting of uniform spaces. Its proof is adapted from [Eng88, p. 440-443; Will70, § 43]. Theorem 3.1.5 is an analogue of the Arzela-Ascoli theorem for the β-topology, as given in [Kh79]. We mention that the abstract version of the Arzela-Ascoli theorem was first given by Myers [My46] which was later improved by Gale [Gal50]. The name k-space also first appeared in the paper [Gal50]. Section 3.2. Here the Arzéla-Ascoli type theorems which characterize precompact subsets of weighted functions spaces CVo (X, E) and CVpc (X, E) are taken from [RS88, KO05]. When the range space E happens to be quasi-complete, these characterizations become criteria for relative compactness. Theorem 3.2.2. is an analogue of Bierstedt [Bie75b], Ruess and Summers ([RS88], Lemma 2.3) and Oubbi ([Ou02], Proposition 9).

CHAPTER 4

The Stone-Weierstrass type Theorems In this chapter, we are primarily concerned with presenting some generalizations of the classical Stone-Weierstrass theorem to Cb (X, E) and CVb (X, E) endowed with various topologies for E a non-locally convex TVS. We also consider Weierstrass polynomial approximation, characterization of maximal submodules and the approximation property for vector-valued functions. Both X and E are always assumed to be Hausdroff..

63

64

4. THE STONE-WEIERSTRASS TYPE THEOREMS

1. Approximation in the Strict Topology Various versions of the Stone-Weierstrass theorem for the spaces (Cb (X, E), β), (Cb (X, E), u) and (C(X, E), k) are considered in this section. Recall that M.H. Stone ([St37], p. 466) obtained the following generalization of the Weierstrass approximation theorem of 1885 (see [St48]; [Simm62], p.157-167). Theorem A. (Classical Stone-Weierstrass theorem for C(X) = C(X,C).) Let X be a compact space. Suppose a subset A of C(X) satisfies (1) A is a self-adjoint subalgebra, (2) A separates points of X, i.e., for any x 6= y in X, there exists an f ∈ A such that f (x) 6= f (y), (3) A does not vanish on X, i.e., for each x ∈ X. there exists a function g ∈ A such that g(x) 6= 0. Then A is u-dense in C(X). Note. If X is locally compact, then the above theorem holds for Co (X) in place of C(X). We now consider the case of vector-valued functions. If E is a vector space, then C(X, E) need not be an algebra but is a Cb (X)-module. An apparent analogue of Theorem A may be stated in the following form. Theorem B. Let X be a compact space and E a normed space (or a TVS). Suppose a subset A of C(X, E) satisfies (i) A is a C(X)-submodule, (ii) A separates points of X, (iii) A does not vanish on X. Then A is u-dense in C(X, E). Unfortunately, Theorem B is not true in general as we now show. Counter-example: [Buc58] For any fixed xo ∈ X and M a proper closed vector subspace of a TVS E, let S(xo , M) = {f ∈ Cb (X, E) : f (xo ) ∈ M}. Then A = S(xo , M) satisfies conditions (i), (ii), (iii) as follows. Clearly, A is a C(X)-submodule of C(X, E). To verify (ii), let x 6= y ∈ X. Since X, being a compact Hausdorff space, is normal, there exists a function ϕ ∈ Cb (X) such that 0 ≤ ϕ ≤ 1, ϕ(x) = 0 and ϕ(y) = 1. For any a(6= 0) in M, (ϕ ⊗ a)(xo ) = ϕ(xo )a ∈ M and so ϕ ⊗ a ∈ A. Further (ϕ ⊗ a)(x) 6= (ϕ ⊗ a)(y). To verify (iii), consider any b(6= 0) in M. If 1 is the unit function in Cb (X), then 1 ⊗ b is a non-zero constant function in

1. APPROXIMATION IN THE STRICT TOPOLOGY

65

A. Clearly, A = S(xo , M) is a proper u-closed subspace of C(X, E) and so cannot be u-dense in C(X, E). Thus one must look for an appropriate replacement of conditions (i), (ii) and (iii) in Theorem B. Kaplansky ([Kap51], §3) had considered similar conditions and obtained some non-commutative generalizations of the classical Stone-Weierstrass theorem, replacing E by a family {Ex : x ∈ X} of C ∗ -algebras having units. But in our setting, the following theorem for the strict topology β on Cb (X, E) was established by R.C. Buck [Buc58]. Theorem 4.1.1. (Stone-Weierstrass theorem for (Cb (X, E), β) ). Let X be a locally compact metrizable space and E a finite dimensional normed space. Suppose a subset A of Cb (X, E) satisfies (1) A is a Cb (X)-submodule, (2) for each x ∈ X, the set A(x) = {g(x) : g ∈ A} is dense in E. Then A is β-dense in Cb (X, E). The above result was later extended to the case of X any locally compact space and E a locally convex TVS by Todd [Tod65] and Wells [Wel65], independently. In this chapter we first present a Stone-Weierstrass type theorem, due to Khan [Kh79], with E any TVS but under the additional condition of finite covering dimension on X (see Section A.1). The following lemma on “partition of unity”, due to Nachbin ([Nac67], Lemma 2, p. 69) plays a crucial role in the proof of Theorem 4.1.3 and also in some later theorems. Lemma 4.1.2. (Partition of Unity) Let X be a completely regular space, and let M ⊆ C(X, K) be a Cb (X, K)-submodule. Let K be a compact subset of X such that ( ∗) M does not vanish on K. Then, given any open cover {U1 , ..., Un } of K, there exist ϕ1 , ..., ϕn ∈ M, such that 0 ≤ ϕi ≤ 1, ϕi = 0 outside Ui ,

n X i=1

ϕi = 1 on K, and

n X

ϕi ≤ 1 on X.

i=1

We mention that if M = Cb (X, K), (∗) clearly holds. Proof. By (∗), for each t ∈ K, there is an ft ∈ M such that ft (t) 6= 0. We may assume that ft ≥ 0. [Indeed, let γ : K −→ K be defined by  if x = 0,  0 2 γ(x) = |x| /x if 0 < |x| < 1,  |x| /x if |x| ≥ 1.

66


Clearly, γ is bounded, continuous and satisfies γ(x) 6= 0 for x 6= 0 and γ(x)x ≥ 0 for x ∈ K. Set φ = γ ◦ ft ∈ Cb (X, K); then φ(x) 6= 0 and φft ≥ 0. We replace ft by φft ∈ M to achieve ft ≥ 0]. We may also assume that ft = 0 outside some Ui . [In fact, choose some Ui containing t and ψ ∈ C(X, K) such that 0 ≤ ψ ≤ 1, ψ(t) = 1 and ψ = 0 outside Ui . Therefore, it would be sufficient to replace ft by ψft ∈ M to assure that ft = 0 outside some Ui .] By compactness of K, there exists a finite collection {f1 , ..., fm } ⊆ {ft : t ∈ K} ⊆ M such that fj ≥ 0 on X, fj = 0 outside someUj (1 ≤ j ≤ m) and

m X

fj (x) > 0

j=1

for allx ∈ K.

For eachP i = 1, ..., n, define Ji = {j : fj = 0 outside Ui , 1 ≤ j ≤ m}, and let gi = j∈Ji fj . Then gi ∈ M, gi ≥ 0, gi = 0 outside Ui (1 ≤ i ≤ n) and

n X

gi (x) > 0 for all x ∈ K.

i=1

ByP compactness of K, there is anP h ∈ Cb (X, K) such that 0 ≤ h ≤ n 1/ i=1 gi on X, and that h = 1/ ni=1 gi on K. Setting ϕi = hgi , we clearly have 0 ≤ ϕi ≤ 1, ϕi = 0 outside Ui ,

n X i=1

ϕi = 1 on K and

n X

ϕi ≤ 1

i=1

on X.

Theorem 4.1.3. [Kh79] Let X be a completely regular space of finite covering dimension and E a TVS. If A is a Cb (X)-submodule of Cb (X, E) such that, for each x ∈ A, A(x) is dense in E, then A is β-dense in Cb (X, E). Proof. Suppose X has covering dimension of order n, and let f ∈ Cb (X, E). Let ϕ ∈ Bo (X), 0 ≤ ϕ ≤ 1, and W ∈ W. There exists a balanced V ∈ W such that V + V + ... + V ((n + 2)-terms) ⊆ W . Let K be a compact subset of X such that ϕ(x)f (x) ∈ V for x ∈ / K. For each x ∈ X, choose a function gx in A and an open neighborhood G(x) of x such that gx (y) − f (y) ∈ V for all y ∈ G(x).


67

The sets in {G(x) : x ∈ K} form an open cover of K, and so there exists a finite open cover, {G(xj ): j = 1, ..., m} say, of K. The sets in U = {X\K, G(xj ) : j = 1, ..., m} form a finite open cover of X, and so, by hypothesis, there exists an open cover B of X such that B is a refinement of U and any point of X belong to at most n + 1 members of B. Since K is compact, a finite number of members of B, U1 , ..., Ur say, will cover K. Moreover, since B is a refinement of U, for each 1 ≤ i ≤ r, there exists a ji , 1 ≤ m, such that Ui ⊆ G(xji ). By Lemma 4.1.2, there exist Pr 0 ≤ ϕi ≤ 1, ϕi = 0 outside of Ui , Pr ϕ1 , ..., ϕr ∈ Cb (X) such that n=1 ϕi (x) ≤ 1 for x ∈ X. We define an n=1 ϕi (x) = 1 for x ∈ K, and E-valued function g on X by g(x) =

r X

ϕi (x)gxji (x),

x ∈ X,

n=1

where gxji is the function in A chosen as indicated earlier. Then g ∈ A. Let y be any point in X. If Iy = {i : y ∈ Ui }, then Iy has at most (n + 1)-members and ϕi (y) = 0 if i ∈ / Iy . Consequently, if y ∈ K, then ( r ) X ϕ(y)(g(y) − f (y)) = ϕ(y) ϕi (x)(gxji (y) − f (y)) = ϕ(y)

n=1 X 

ϕi (y)(gxji (y) − f (y))

i∈Iy

  

∈ V + V + ... + V (at most (n+1)-times) ⊆ W. If y ∈ / K, we have ϕ(y)(g(y) − f (y)) = ϕ(y) +

r X

i=1 ( r X

ϕi (y)[gxji (y) − f (y)]) )

ϕi (y) − 1 ϕ(y)f (y)

i=1

(∗)

∈ V + ... + V (at most (n+1)-times) + V ⊆ W. Thus g − f ∈ N(ϕ, W ), and so f belongs to the β-closure of A; that is, A is β-dense in Cb (X, E), as required. An equivalent version of Theorem 4.1.3 is as follows.

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Theorem 4.1.4. [Buc58, Kh79] Let X and E be as in Theorem 4.1.3, and let A be a Cb (X)-submodule of Cb (X, E) and f ∈ Cb (X, E). Then the following conditions are equivalent: (1) f belongs to the β-closure of A. (2) For each x ∈ X, f (x) ∈ A(x). Proof. (1) ⇒ (2) Suppose that f belongs to the β-closure of A, and let x be any point in X. Let {fα } be a net in A such that f = β-limα fα . Since p ≤ β, f = p-limα fα , and so, in particular, fα (x) −→ f (x). Since {fα (x)} ⊆ A(x), it follows that f (x) ∈ A(x). (2) ⇒ (1) Suppose that, for each x ∈ X, f (x) ∈ A(x). If N(ϕ, W ) is any β-neighborhood of 0 in Cb (X, E), then, in the same way as in Theorem 4.1.3, we can find a function h in A such that h−f ∈ N(ϕ, W ). Thus h belongs to the β-closure of A. We next state two versions of the Stone-Weierstrass theorem for the u and k topologies. Their proofs are omitted as they are similar to that of Theorem 4.1.3. Theorem 4.1.5. [Kh80] Let X be a locally compact space of finite covering dimension and E a Hausdorff TVS. If A is a Co (X)-submodule of Co (X, E) such that, for each x ∈ X, A(x) is dense in E, then A is u-dense in Co (X, E). Theorem 4.1.6. [Kh95] Let X be a completely regular space such that every compact subset of X has a finite covering dimension, and let E be a TVS. If A is a C(X)-submodule of C(X, E) such that, for each x ∈ X, A(x) is dense in E, then A is k-dense in C(X, E). Corollary 4.1.7. Let X be a completely regular space and E a TVS. (a) If X has finite covering dimension, then Cb (X) ⊗ E is β-dense in Cb (X, E). (b) If X is locally compact and has finite covering dimension, then Co (X) ⊗ E is u-dense in Co (X, E). (c) If each compact subset of X has finite covering dimension, then C(X) ⊗ E is E is k-dense in C(X, E). Proof. (a) Since Cb (X) contains the unit function 1, it is easy to see that (Cb (X) ⊗ E)(x) = E for each x ∈ X. It follows from the Theorem 4.1.3 that Cb (X) ⊗ E us β-dense in Cb (X, E). (b) Its proof is similar to part (a) by using Theorem 4.1.5. (c) Its proof is similar to part (a) by using Theorem 4.1.6. Remark 4.1.8. If E is a locally convex TVS, then, with slight modification in the proofs, Theorems 4.1.3-4.1.6 and Corollary 4.1.7 hold without the hypothesis of finite covering dimension.


69

Let Cpc (X, E) (resp. Crc (X, E)) denote the subspace of Cb (X, E) consisting of those functions f such that f (X) is precompact (resp. relatively compact). Note that if E is complete, then Cpc (X, E) = Crc (X, E). Recall that a TVS E is said to have the approximation property if the identity map on E can be approximated uniformly on precompact sets by continuous and linear maps of finite rank; E is said to be admissible if the identity map on E can be approximated uniformly on compact sets by continuous maps of finite rank. Every locally convex TVS and every F -space with a basis (e.g. ℓp , 0 < p < 1) is admissible. Theorem 4.1.9. [Shu72, Kh79, Kh95] Let X be a completely regular space and E a TVS. (a) E is admissible iff for all topological spaces X, Cb (X) ⊗ E is u-dense in Crc (X, E). (b) If E has the approximation property, then Cb (X) ⊗ E is u-dense in Cpc (X, E). (c) If X is a normal space of finite covering dimension and E any TVS, then Cb (X) ⊗ E is u-dense in Crc (X, E). (d) If E is an admissible TVS, then (i) C(X) ⊗ E is k-dense in C(X, E). (ii) Cb (X) ⊗ E is β-dense in Cb (X, E). Proof. (a) Suppose E is admissible, and let f ∈ Crc (X, E) and W ∈ W a balanced set. Since f (X) is compact in E, there exists a continuous map u : f (X) → E with range contained in a finite dimensional subspace of E such that u(a)−a ∈ W for all a ∈ f (X). Then h = u◦f ∈ Cb (X)⊗E and h(x) − f (x) ∈ W for all x ∈ X. Conversely, let K ⊆ E be a compact set and W ∈ W. Since, by hypothesis, Cb (K) ⊗ E is u-dense in Cb (K, E), there exists some v = P n i=1 vi ⊗ai (vi ∈ Cb (K), ai ∈ E) in Cb (K)⊗E such that v(a)−a ∈ W for all a ∈ K. Note that the range of v is contained in the finite dimensional subspace spanned by {a1 , ..., an }. Thus E is admissible. (b) Suppose E has the approximation property, and let f ∈ Cpc (X, E) and W ∈ W a balanced set. Since f (X) is precompact in E, there exists a continuous linear map u : f (X) → E with range contained in a finite dimensional subspace of E such that u(a) − a ∈ W for all a ∈ f (X). Then h = u ◦ f ∈ Cb (X) ⊗ E and h(x) − f (x) ∈ W for all x ∈ X. (c) Since X is normal, its Stone-Cech compactification βX also has finite covering dimension [GJ60]. So, by Corollary 4.1.7, Cb (βX) ⊗ E is u-dense inCb (βX, E). Note that each function in Crc (X, E) has a continuous extension to all of βX. Hence Cb (X) ⊗ E and Crc (X, E) are

70


linearly isomorphic to Cb (βX) ⊗ E and Cb (βX, E), respectively, and so the result follows. (d) (i) Let f ∈ C(X, E), and let K be a compact subset of X and W ∈ W a balanced set. By hypothesis, there exists a continuous map u : f (K) → E with range contained in a finite dimensional subspace of E such that u(f (x)) − f (x) ∈ W for all x ∈ K. We can write u◦f =

m X

(ui ◦ f ) ⊗ ai , where ui ◦ f ∈ C(K) and ai ∈ E.

i=1

By the famous Tietze extension Theorem, Pm there exist h1 , ..., hm ∈ C(X) such that hi = ui ◦ f on K. Let g = i=1 hi ⊗ ai . Then g ∈ C(X) ⊗ E and g − f ∈ N(K, W ). (d) (ii) Let f ∈ Cb (X, E), and let ϕ ∈ Bo (X) and W ∈ W. Choose an open balanced neighborhood V ∈ W such that V + V + V ⊆ W . Choose r > kϕk with f (x) ⊆ rV , and let K = {x ∈ X : ϕ(x) ≥ 1/r}. Then f (K) is a compact subset of E and so, by hypothesis, there exists a continuous map u : f (K) → E with range contained in a finite dimensional subspace of E such that u(f (x)) − f (x) ∈ (1/r)G for all x ∈ K. We can write u◦f =

n X

(ui ◦ f )(x)ai (ui ◦ f ∈ C(K) and ai ∈ E).

i=1

Again Pthere exist θi (1 ≤ i ≤ n) in Cb (X) such that θi = ui ◦ f on K. Let h = ni=1 θi ⊗ ai . Then K ⊆ h−1 (rV + rV ) = F (say),

which is open in X, and so there exists a θ ∈ Cb (X) with 0 ≤ θ ≤ 1, θ = 0 on X\F . Let g = θh. Then g ∈ Cb (X) ⊗ E and g = h = u ◦ f on K. Further, g(x) ⊆ rV + rV . It is now easily verified that g − f ∈ N(ϕ, W ). Next, we give a brief account of an interesting and more general version of the Stone-Weierstrass theorem due to Prolla [Pro93b]. Most of the preceding versions of this theorem describe the closures of “Cb (X)submodules” of Cb (X, E) under various topologies. Prolla [Pro93b] has established a result that describes the u-closure of certain “subsets” of Cb (X, E) which are not necessarily linear subspaces (see also [Pro93a,


71

PN97]). Its proof uses two results of Jewitt ([Je63], Lemma 2 and Theorem 1) and a “lemma” of Prolla ([Pro93b], Lemma 3) on “approximating the characteristic functions of open neighborhoods”. (This lemma is an analogue of Lemma 1.3 of Machado’s proof [Mac77] of the StoneWeierstrass theorem, and of Lemma 1 of Brosowski and Deutsch [BD81]. We shall not reproduce the proofs here since they can also be found in the book ([Pro93a], Chapter 11). We first introduce some notations and terminology. Let X be a compact space and E a TVS. (In this case C(X, E) = Cb (X, E) and β = u.) Let I = [0, 1], and let C(X, I) denote the subset of C(X, R) consisting of those functions from X into I. A subset M of C(X, I) is said to have property (V ) [Je63] if (1) ϕ ∈ M implies 1 − ϕ ∈ M; (2) ϕ, θ ∈ M implies ϕθ ∈ M. If A ⊆ C(X, E), then a function ϕ ∈ C(X, I) is called a convex multiplier of A if ϕf + (1 − ϕ)g ∈ A for every pair f, g ∈ A. Let MA denote the set of all multipliers of a subset A of C(X, E). Clearly, ϕ ∈ MA implies 1 − ϕ ∈ MA . Further, if ϕ, θ ∈ M, then the identity (ϕθ)f + (1 − ϕθ)g = ϕ(θf + (1 − θ)g] + (1 − ϕ)g for f, g ∈ A shows that ϕθ ∈ MA . Hence MA has property V . We mention that property (V ) is considered as a hypothesis in Theorem 1 of [Je63]. The main results of [Pro93a] are stated as follows. Theorem 4.1.10. Let X be a compact space of finite covering dimension and E a TVS. Let A be a subset of C(X, E) such that set MA of all multipliers of A separates the points of X. Let f ∈ C(X, E) be given. The following are equivalent: (1) f belongs to u-closure of A. (2) For each x ∈ X, f (x) ∈ A(x). Corollary 4.1.11. Let X and E be as given above, and let A be a subset of C(X, E) such that (a) the set MA of multipliers separates points of X; (b) for each x ∈ X, A(x) is dense in E. Then A is u-dense in C(X, E). Corollary 4.1.12. Let X and E be as given above. Then C(X) ⊗ E is u-dense in C(X, E).

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2. Approximation in the Weighted Topology We next consider the weighted approximation problem. Let V b a Nachbin family on X, M ⊆ C(X) a Cb (X)-submodule, and A ⊆ CVo (X, E) an M-submodule. The ‘weighted approximation problem’, formulated originally by Nachbin [Nac65], consists in finding a characterization of the ωV -closure of A in CVo (X, E). For M ⊆ C(X) or C(X, E), we shall denote z(M) = {x ∈ X : g(x) = 0 for all g ∈ M}. Theorem 4.2.1. [Pr71a, Kh85] Let X be a completely regular space of finite covering dimension, E a TVS, V > 0 a Nachbin family on X, and M ⊆ C(X) a Cb (X)-submodule. Let A ⊆ CVo (X, E) be an Msubmodule such that z(M) ⊆ z(A), and let f ∈ CVo (X, E). Then f belongs to the ωV -closure of A iff, for each x ∈ X, f (x) ∈ A(x). Proof. (⇒) Suppose f belongs to the ωV -closure of A. Since V > 0, ρ ≤ ωV and so f belongs to the ρ-closure of A; that is, or each x ∈ X, f (x) ∈ A(x). (⇐) Suppose X has covering dimension n, and let v ∈ V and G ∈ W. Choose a balanced H ∈ W with H + H + ... + H((n + 2)-terms) ⊆ G. Then K = {x ∈ X : v(x)f (x) ∈ / H} is compact. By hypothesis, for each x ∈ K, we can choose a gx ∈ A such that f (x) − gx (x) ∈ H. Note that gx (x) 6= 0 and so, since z(M) ⊆ z(A), K ⊆ X\z(M). Choose open neighborhoods S ′ (x) and S ′′ (x) of x in X such that v(y) ≤ 1 for all y ∈ S ′ (x) and f (y) − gx (y) ∈ H for all y ∈ S ′′ (x). Put S(x) = S ′ (x) ∩ S ′′ (x). Then open cover {S(x) : x ∈ K} has a finite subcover {S(xj ) : 1 ≤ j ≤ m} (say). Let U be an open refinement of {X\K, S(xj ) : 1 ≤ j ≤ m} such that any point of X belongs to at most n + 1 members of U. Choose {Ti : 1 ≤ i ≤ r} ⊆ U which covers K. Since K ⊆ X\z(M), we can apply Lemma 4.1.2 to obtain a collection {ϕi : 1 ≤ i ≤ r} ⊆ MPsuch that 0 ≤ ϕi ≤ 1, ϕi = P0r outside Ti , P r r ϕ ≤ 1 on X, Then g = ϕ = 1 on K, and i=1 ϕi gji ∈ A i=1 i i=1 i and, for any y ∈ X, v(y)[g(y) − f (y)] =v(y)

r X i=1

ϕi (y)[gji (y) − f (y)] + [

r X

ϕi (y) − 1]v(y)f (y)

i=1

∈H + ... + H (at most (n+1)-terms) + H ⊆ G

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(by considering the cases y ∈ K and y ∈ / K); that is, g − f ∈ N(v, G). Thus f belongs to the ωV -closure of A. Corollary 4.2.2. Let X, E, V and A be as in Theorem 4.2.1 and M = Cb (X). Then A is ωV -dense in CVo (X, E) iff for each x ∈ X, A(x) is dense in E. In particular, CVo (X) ⊗ E is ωV -dense in CVo (X, E). Theorem 4.2.3. [Kh87] (a) CVo (X, E) is ωV -closed in CVb (X, E). (b) If X is locally compact, then Coo (X, E) is ωV -dense in CVo (X, E). Proof. (a) Let f belong to the ωV -closure of CVo (X, E) in CVb (X, E). To show that f ∈ CVo (X, E), let v ∈ V . Then, for any W ∈ W, the set F = {x ∈ X: v(x)f (x) ∈ / W } is compact, as follows. Choose a balanced G ∈ W with G + G ⊆ W . There exists a function g ∈ CVo (X, E) such that g − f ∈ N(v, G). Now the set K = {x ∈ X : v(x)g(x) ∈ / G} is compact. If x ∈ X\K, then v(x)f (x) ∈ v(x)g(x) + G ⊆ G + G ⊆ W, and so x ∈ X\F. Therefore, F ⊆ K which implies that F is compact. Thus f ∈ CVo (X, E). (b) Let f ∈ CVo (X, E). To show that f belongs to the ωV -closure of Coo (X, E), let v ∈ V and W ∈ W be balanced. Now K = {x ∈ X : v(x)f (x) ∈ / W } is compact. Since X is locally compact, there exists a ψ ∈ Coo (X) such that 0 ≤ ψ ≤ 1 and ψ = 1 on K. Then clearly g = ψf ∈ Coo (X, E). Let x ∈ X. If x ∈ K, then v(x)[f (x) − g(x)] = v(x)[f (x) − f (x)] = 0 ∈ W ; If x ∈ X\K, then v(x)[f (x) − g(x)] = [ψ(x) − 1]v(x)f (x) ∈ [ψ(x) − 1]W ⊆ W ; that is g − f ∈ N(v, W ). Thus Coo (X, E) is ωV -dense in CVo (X, E). Theorem 4.2.4. [Kh87] Let X be a completely regular Hausdorff space. Suppose E is a locally bounded TVS. Then Cb (X, E) ∩ CVo (X, E) is ωV -dense in CVo (X, E). Proof. Let B be a bounded neighborhood of 0 in E. There exists a closed shrinkable neighborhood S of 0 in E with S ⊆ B. The Minkowski functional ρ = ρS of S is continuous and positively homogeneous, and S = {a ∈ E : ρ(a) ≤ 1}.Consequently, for any t > 0, the function

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ht : E −→ E defined by ht (a) =

a t a ρ(a)

if a ∈ tS, if a ∈ / tS,

is continuous. Further, ht (E) ⊆ tS. [Let a ∈ E. If a ∈ tS, clearly 1 ht (a) = a ∈ tS; if a ∈ / tS, then ρ(a) > t > 0 and ρ(a) a ∈ S, and so t ht (a) = ρ(a) a ∈ tS.] Now, for any f ∈ CVo (X, E), let Ft = ht ◦ f. We claim that Ft ∈ Cb (X, E) ∩ CVo (X, E). Clearly Ft is continuous on X; further, Ft is bounded since Ft (X) = ht (f (X)) ⊆ ht (E) ⊆ tS ⊆ tB and B is bounded. Hence Ft ∈ Cb (X, E). Now, let v ∈ V. To show that vFt vanishes at infinity, we need to show that, for any balanced G ∈ W, the set A = {x ∈ X : v(x)Ft (x) ∈ / G} is compact. [Since f ∈ CVo (X, E), the set B = {x ∈ X : v(x)f (x) ∈ / G} is compact. Let x ∈ X\B, so that v(x)f (x) ∈ G. Then, if f (x) ∈ tS, v(x)Ft (x) = v(x)ht (f (x)) = v(x)f (x) ∈ G and if f (x) ∈ / tS, ρ(f (x)) > t and v(x)Ft (x) = v(x)ht (f (x)) =

t t v(x)f (x) ∈ G ⊆ G; ρ(f (x)) ρ(f (x))

that is, x ∈ X\A. Therefore A ⊆ B, which implies that A is compact.] Finally, let f ∈ CVo (X, E), and let v ∈ V and W ∈ W be balanced. We show that, for a suitable t > 0, the function Ft = ht ◦ f. satisfies Ft − f ∈ N(v, W ). [Choose r ≥ 1 such that B ⊆ rW and B ⊆ rS. The set K = {x ∈ X : v(x)f (x) ∈ / 1r S} is compact and so we choose t ≥ 1 with f (K) ⊆ rt B. Now, let x ∈ X. If f (x) ⊆ tS, then v(x)[Ft (x) − f (x)] = v(x)[ht (f (x)) − f (x)] = v(x)[f (x) − f (x)] = 0 ∈ W. If f (x) ∈ / tS, then ρ(f (x)) > t and also x ∈ / K (since f (K) ⊆ rt B ⊆ tS) and so v(x)f (x) ∈ 1r S; hence t t f (x) − f (x)] = [ − 1]v(x)f (x) ρ(f (x)) ρ(f (x)) 1 t t − 1]( )S ⊆ [ − 1]W ⊆ W. ∈ [ ρ(f (x)) r ρ(f (x))

v(x)[Ft (x) − f (x)] = v(x)[

Thus Ft − f ∈ N(v, W ).]

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Theorem 4.2.5. [Kh87] Let X be a completely regular space. Suppose E is an admissible TVS and V ⊆ So+ (X). Then Cb (X) ⊗ E is ωV -dense in CVo (X, E). Proof. Let f ∈ Cb (X, E), and let v ∈ V and W ∈ W. Choose an open balanced G ∈ W such that G + G + G ⊆ W . Choose r ≥ max{1, kvk} with f (X) ⊆ rG, and let K = {x ∈ X : v(x) ≥ 1r }. Then f (K) is a compact subset of E and so, by hypothesis, there exists a continuous map ϕ : f (K) −→ E with range contained in a finite dimensional subspace of E such ϕ(f (x)) − f (x) ∈ 1r G for all x ∈ K. We can write Pthat n ϕ ◦ f = i=1 (ϕi ◦ f ) ⊗ ai , where ϕi ◦ f ∈ C(K) and ai ∈ E. By the Tietze extension Theorem, therePexist ψi (1 ≤ ψi ≤ 1) in Cb (X) such that ψi = ϕi ◦ f on K. Let h = ni=1 ψi ⊗ ai . For any x ∈ K, 1 h(x) = (ϕ ◦ f )(x) ∈ f (x) + G ⊆ rG + rG; r −1 hence K ⊆ h (rG + rG) = F (say). Clearly F is open in X, and so there exists a ψ ∈ Cb (X) with 0 ≤ ψ ≤ 1, ψ = 1 on K and ψ = 0 on X\F. Let g = ψh. Then g ∈ Cb (X) ⊗ E and g = h = ϕ ◦ f on K. Further, g(X) ⊆ rG + rG. [Let x ∈ X. If x ∈ F, then g(x) = ψ(x)h(x) ∈ ψ(x)(rG + rG) ⊆ (rG + rG); if x ∈ / F, then g(x) = 0 ∈ rG + rG.] Finally, g − f ∈ N(v, W ), as follows. Let x ∈ X. If x ∈ K, then 1 v(x)[g(x) − f (x)] = v(x)[ϕ(f (x)) − f (x)] ∈ v(x) G ⊆ G ⊆ W ; r 1 if x ∈ / K, then v(x) < r and so v(x)[g(x) − f (x)] ∈ v(x)[rG + rG − rG] ⊆ v(x)rW ⊆ W. Corollary 4.2.6. If E is locally bounded and admissible and V ⊆ then Cb (X) ⊗ E is ωV -dense in CVo (X, E). We do not know whether, for any Nachbin family V and E admissible, CVo (X) ⊗E is ωV -dense in CVo (X, E). However, under some restrictions on X, this is true for E any TVS. So+ (X),

Theorem 4.2.7. [Kh87] Let X be a locally compact space of finite covering dimension. Then Coo (X) ⊗ E is ωV -dense in CVo (X, E). Proof. By Theorem 4.2.3 (b), Coo (X, E) is ωV -dense in CVo (X, E). Therefore, it suffices to show that Coo (X) ⊗ E is ωV -dense in Coo (X, E). Let f ∈ Coo (X, E), and let v ∈ V and W ∈ W be balanced. There exists

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a compact set K ⊆ X such that f (x) = 0 for x ∈ / K. Choose r ≥ 1 with v(x) ≤ r for all x ∈ K. Since X is locally compact and E is of finite covering dimension, by Corollary 4.1.7, Co (X) ⊗ E is u-dense in Co (X, E); hence Coo (X) ⊗ E is u-dense in Coo (X, E). Then there exists a function g ∈ Coo (X) ⊗ E with g = 0 outside K and such that 1 g(x) − f (x) ∈ W for all x ∈ X. r Now, let x ∈ X. If x ∈ K, then v(x) v(x)[g(x) − f (x)] ∈ W ⊆ W; r if x ∈ / K, then v(x)[g(x) − f (x)] = v(x)[0 − 0] = 0 ∈ W. Thus g − f ∈ N(v, W ).

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3. Weierstrass Polynomial Approximation In this section, we present an infinite dimensional version of the Weierstrass polynomial approximation theorem, due to Bruno [Bru84]. Let E and F be real TVSs and C(E, F ) the space of all continuous functions from E into F , and let P (E, F ) = span{(ϕ(x))n y : ϕ ∈ E ∗ , x ∈ X, y ∈ F, n ≥ 1}, the set of all continuous polynomials of finite type from E into F . Definition. A sequence {Sn } of continuous linear operators from a TVS E into itself is said to have the Grothendieck approximation property (GAP) if it satisfies: (i) each Sn has finite rank, (ii) {Sn } is equicontinuous, (iii) {Sn } converges uniformly on compact subsets to the identity on E. For example, in a complete metric TVS with a Schauder basis {en }, the sequence of projections ∞ X Sn : ai ei → an en i=1

has the GAP.

Example. For 0 < p < 1, the spaces ℓp (R), given by ℓp (R) = {a = {ai } ⊆ R: P∞

∞ X

|ai |p < ∞},

i=1

p

with metric d(a, b) = i=1 |ai − bi | , are complete metric vector spaces with Schauder bases and are not locally convex [Simo65]. Theorem 4.3.1. [Bru84] Let E and F be real TVS. Suppose E has a sequence {Sn } of projections with the GAP and F has a sequence {Tn } with the GAP. Then the space P (E, F ) of continuous polynomials from E into F of finite rank is k-dense in C(E, F ). Proof. The proof is divided into three steps. k

Step I. We show that, for any f ∈ C(E, F ) and n ≥ 1, Tn f Sn −→ f . Let K ⊆ E be a compact set and W a neighborhood of 0 in F . Choose a balanced neighborhood V of 0 in F with V + V ⊆ W . Since {Tn } is equicontinuous, there exists a neighborhood U of 0 in E such that Tn (U) ⊆ V

for all n ≥ 1.

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Since f is uniformly continuous on K and {Sn } converges uniformly to the identity of E on K, it easily follows that {f Sn } converges uniformly to f on K. So there exists an integer n1 such that f Sn (x) − f (x) ∈ U for all x ∈ K and n ≥ n1 . Since {Tn } converges uniformly to the identity of F on f (K), there exists an integer n2 such that Tn f (x) − f (x) ∈ V for all x ∈ K and n ≥ n2 . Hence, if n ≥ max{n1 , n2 } and x ∈ K, Tn f Sn (x) − f (x) = Tn (f Sn (x) − f (x)) + (Tn f (x) − f (x)) ∈ Tn (U) + V ⊆ V + V ⊆ W . Step II. We show that, for any f ∈ C(E, F ) each Tn f Sn belongs to the k-closure of P (E, F ). Put fn = Tn f Sn . Let K ⊆ E be a compact set and V a balanced neighborhood of 0 in F . Since E and F are Hausdorff T V Ss and Sn , Tn are of finite rank, Sn (E) and Tn (F ) are linearly homeomorphic to finite-dimensional Euclidean spaces. By the classical Stone-Weierstrass theorem [St48], P (Sn (E), Tn (F )) is k-dense in C(Sn (E), Tn (F )). Since fn′ = fn |Sn (E) belong to C(Sn (E), Tn (F )) and Sn (K) is compact in Sn (E), there exists a p′n ∈ P (Sn (E), Tn (F )) such that fn′ (x) − p′n (x) ∈ V for all x ∈ Sn (K). We define an extension pn of p′n (x) from Sn (K) to E by pn (x) = p′n Sn (x), x ∈ E. Clearly pn ∈ P (E, F ) and for any x ∈ K, fn (x) − pn (x) = fn Sn (x) − p′n Sn (x) ∈ V. Step III. We now verify that P (E, F ) is k-dense in C(E, F ). Let f ∈ C(E, F ), and let K ⊆ E be compact and W a neighborhood of 0 in F . Choose a balanced neighborhood V of 0 in F with V + V ⊆ W . By step I, there exists an integer n such that f (x) − Tn f Sn (x) ∈ V for all x ∈ K. By step II, there exists a pn ∈ P (E, F ) such that Tn f Sn (x) − pn (x) for all x ∈ K. Hence, for any x ∈ K, f (x) − pn (x) = [f (x) − Tn f Sn (x)] + [Tn f Sn (x) − pn (x)] ∈ V + V ⊆ W.

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4. Maximal Submodules in (Cb (X, E), β) This section includes characterizations of maximal and closed Cb (X)submodules of (Cb (X, E), β) and CVo (X, E),as given in [Kh84, Kh85b]. Throughout this section, X will denote a completely regular Hausdorff space and E a Hausdorff TVS with non-trivial topological dual E ∗ .

Definition. A TVS F is said to have the separation property (or, equivalently, the Hahn-Banach extension property) if, for any closed vector subspace S of F and a ∈ F \S, there exists a maximal (proper) closed vector subspace M of F such that S ⊆ M and a ∈ / M. F is said to have the weaker separation property if every proper closed vector subspace of F is contained in some maximal closed subspace of F . Clearly, the separation property implies the weaker separation property. Every locally convex TVS has the separation property [Scha66], and it is shown by Gregory and Shapiro [GrSh70] that if hF, F ′i is a dual pair of a vector space F with the weak topology on F not equal to the Mackey topology, then there is a non-locally convex linear topology between them having the separation property. We shall require the following version of the Stone-Weierstrass theorem. Theorem 4.4.1. [Kh84] For any Cb (X)-submodule A of Cb (X, E), consider the following conditions. (1) A is β-dense in Cb (X, E). (2) For each x ∈ X, A(x) is dense in E. (3) For each x ∈ X and maximal closed subspace M of E, there exists an f ∈ A such that f (x) ∈ / M. Then (1) ⇒ (2) ⇒ (3). If X has finite covering dimension, then (2) ⇒ (1); if E has the weaker separation property, then (3) ⇒ (2). Proof. (1) ⇒ (2) is evident since ρ ≤ β. (2) ⇒ (1) is same as Theorem 4.1.3. (2) ⇒ (3). Let x ∈ X and M a maximal closed subspace of E, and let a ∈ E\M. Choose an open neighborhood G of a in E such that G ⊆ E\M. Since A(x) is dense in E, there exists an f ∈ A with f (x) ∈ G. Hence f (x) ∈ / M. (3) ⇒ (2). Suppose E has the weaker separation property. If A(x) 6= E for some x ∈ X, there exists a maximal closed subspace M of E such / M, and that A(x) ⊆ M. But, by (3), there exists a g ∈ A with g(x) ∈ so A(x) ⊆ M. This gives us the desired contradiction.

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Corollary 4.4.2. [Kh84] Suppose X has finite covering dimension and E has the separation property. Let A be a Cb (X)-submodules of Cb (X, E), and f ∈ Cb (X, E). Then the following are equivalent: (1) f belongs to the β-closure of A. (2) For each x ∈ X, f (x) ∈ A(x). (3) For each x ∈ X and maximal closed subspace M of E, A(x) ⊆ M implies that f (x) ∈ M. We now characterize the maximal β-closed Cb (X)-submodules of Cb (X, E). We begin with the following. Lemma 4.4.3. [Kh84] For any x ∈ X and M a maximal closed subspace of E, S(x, M) = {f ∈ Cb (X, E) : f (x) ∈ M} is a maximal β-closed Cb (X)-submodule of Cb (X, E). Proof. First note that, since M is a vector subspace, S(x, M) is a Cb (X)-submodule of Cb (X, E). There exists a non-zero ϕ ∈ E ∗ such that M = ϕ−1 (0) (§ A.2) Given any x ∈ X, define a map Tx : (Cb (X, E), β) → E by Tx (f ) = f (x), f ∈ Cb (X, E). If W is any neighborhood of 0 in E, then N({x)}, W ) = {f ∈ Cb (X, E) : f (x) ∈ W } is a β-neighborhood of 0 in Cb (X, E) such that Tx (N({x}, W )) ⊆ W. Hence Tx is continuous. It is easy to see that S(x, M) = (ϕ ◦ Tx )−1 (0), and so S(x, M) is maximal and β-closed.

Theorem 4.4.4.[Kh84] Suppose X has finite covering dimension. (1) If E has the weaker separation property, then every maximal β-closed Cb (X)-submodule of Cb (X, E) is of the form S(x, M), where x ∈ X and M is a maximal closed subspace of E. (2) If E has the separation property, then every β-closed Cb (X)submodule of Cb (X, E) is the intersection of all maximal β-closed Cb (X)submodules which contain it. Proof. (1) Let A be a maximal β-closed Cb (X)-submodule of Cb (X, E), and let f ∈ Cb (X, E) with f ∈ / A. Since A is β-closed, by Theorem 4.4.1, there exists an xo ∈ X such that f (xo ) ∈ / A(xo ). By the weaker separation property of E, there exists a maximal closed subspace M of E

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81

such that A(xo ) ⊆ M. Clearly, A ⊆ S(xo , M). Hence, by Lemma 5.5.3, A = S(xo , M). (2) Let A1 be a β-closed Cb (X)-submodule of Cb (X, E), and let g ∈ Cb (X, E) with g ∈ / A1. Then, as in the above proof and using Corollary 4.4.2, there exist an x1 ∈ X and a maximal closed subspace M1 of E such that A ⊆ S(x1 , M1 ) and g ∈ / S(x1 , M1 ).

Remarks. (1) If E is assumed to be locally convex, then Theorems 4.4.1 and 4.4.4, hold without restricting X to have finite covering dimension (see Remark 4.1.8). (2) In general, a maximal Cb (X)-submodule of Cb (X.E) need not be β-closed. For instance if E admits a discontinuous linear functional ϕ and M = ϕ−1 (0), then, for any x ∈ X, S(x, M) is maximal but not β-closed by Theorem 4.4.1 (since M is not closed in E). Finally, we give a characterization of the maximal ωV -closed Cb (X)submodules of CVo (X, E). Theorem 4.4.5. [Kh85b] Let X be of finite covering dimension, E a TVS, and V > 0 a Nachbin family on X. Suppose that E has the separation property. Then: (a) Every maximal ωV -closed Cb (X)-submodule of CVo (X, E) is of the form SV (x, M) = {f ∈ CVo (X, E) : f (x) ∈ M}, where x ∈ X and M is a maximal closed subspace of E. (b) Every ωV -closed Cb (X)-submodule of CVo (X, E) is the intersection of all maximal ωV -closed Cb (X)-submodules which contain it. Proof. (a) Let A be a maximal ωV -closed Cb (X)-submodule of CVo (X, E), and let f ∈ CVo (X, E) with f ∈ / A. By Theorem 4.2.1, there exists an x ∈ X such that f (x) ∈ / A(x). By the separation property of E, there exists a maximal closed subspace M of E such that A(x) ⊆ M and f (x) ∈ / M. It can be easily shown that S(x, M) is a maximal ωV -closed Cb (X)submodule of CVo (X, E), and thus A = SV (x, M). (b) The proof is similar to that of the above part and Theorem 4.4.4 (2).

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5. The Approximation Property Recall that a TVS E is said to have the approximation property if the identity map on E can be approximated uniformly on precompact sets by continuous linear maps of finite rank (i.e. with range contained in finite dimensional subspaces of E). Theorem 4.5.1. [Kat81] Let X be a completely regular space of finite covering dimension and E a Hausdorff TVS having the approximation property. Then (Cb (X, E), β) has the approximation property. Proof. Suppose X has finite covering dimension n. Let A be a βprecompact subset of Cb (X, E), and let ϕ ∈ Bo (X) with 0 ≤ ϕ ≤ 1 and W ∈ W a balanced set. We show that there exists a continuous linear mapping T : (Cb (X, E), β) → (Cb (X, E), β) of finite rank such that T f − f ∈ N(ϕ, W ) for all f ∈ A.

(1)

Choose a balanced W1 ∈ W with W1 + ... + W1 (n + 2 terms) ⊆ W . Also choose an open balanced W2 ∈ W with W2 + W2 + W2 + W2 ⊆ W1 . Since A, being β-bounded, is u-bounded, there exists an r ≥ 1 such that f (X) ⊆ rW2 for all f ∈ A. Choose a compact K ⊆ X such that ϕ(x) < 1/r for all x ∈ X\K. Put B = ∪{f (K) : f ∈ A}. It is easy to see that B is a precompact subset of E. Since E has the approximation property, there exists a continuous linear mapping S : E → E of finite rank such that sa − a ∈ (1/r)W2 for all a ∈ B. (2) Choose an open balanced W3 ∈ W such that W3 ⊆ W2 ∩ S −1 (W2 ) = W2 ∩ {a ∈ E : Sa ∈ W2 }.

(3)

Since A is β-precompact, there exists a finite set A1 ⊆ A such that For each x ∈ X, the set G(x) =

A ⊆ A1 + N(ϕ, W3 ).

(4)

\

(5)

(y ∈ X : g(y) − g(y) ∈ W3 }

g∈A1

is an open neighborhood of x. Since K is compact, there exist x1 , ..., xp ∈ K such that K ⊆ ∪pi=1 G(xi ). The collection {G(x1 ), ..., G(xp ), X\K} forms an open cover of X. Since X has finite covering dimension n and K is compact, there exists an open cover {H1 , ..., Hm } of K which is a refinement of the cover {G(x1 ), ..., G(xp )} and such that each point of K belongs to at most n + 1 of the Hi ’s. For each Hi , choose an

5. THE APPROXIMATION PROPERTY

83

index ji such that Hi ⊆ G(yi ), yi = xji . By Lemma 4.1.2, there exist h1 , ..., hm ∈ Cb (X) with 0 ≤ hi ≤ 1, hi = 0 on X\Hi , P and m i=1 hi (x) ≤ 1 for each x ∈ X. Define T : Cb (X, E) → Cb (X, E) by T (f ) =

m X

m X

hi (x) = 1 on K,

i=1

hi ⊗ Sf (yi ), f ∈ Cb (X, E).

i=1

It is easy to verify that T is a β-continuous linear mapping of finite rank. To complete the proof, we need to verify (1). Let f ∈ A, and let x ∈ X. By (4), there exists a g ∈ A such that ϕ(f − g)(X) ⊆ W3 .

(6)

Note that Ix = {i : x ∈ Hi } contains at most n + 1 indices. Now T f (x) − f (x) =

m X

hi (x)Sf (yi ) − f (x).

i=1

For any i ∈ Ix , x ∈ Hi ⊆ G(yi ) and so, using (2), (3) and (6) ϕ(x)[Sf (yi ) − f (x)] = S[ϕ(x)(f (yi ) − g(yi )] + ϕ(x)[Sg(yi ) − g(yi )] +S[ϕ(x)(g(yi ) − g(x)] + ϕ(x)[g(x) − f (x)] ∈ S(W3 ) + (ϕ(x)/r)W2 + ϕ(x)W3 + W3 ⊆ W2 + W2 + W2 + W2 ⊆ W1 . Pm Case I. Suppose x ∈ ∪m i=1 Hi . If x ∈ K, then i=1 hi (x) = 1 and so " m # m X X ϕ(x)[T f (x) − f (x)] = ϕ(x) hi (x)Sf (yi ) − hi (x))f (x) i=1

= ∈

m X

i=1 m X i=1

i=1

hi (x)ϕ(x)[Sf (yi ) − f (x)] hi (x)W1 ⊆ W1 + ... + W1 ⊆ W. (n+1 terms)

84


If x ∈ (∪m i=1 Hi )\K, then

Pn

hi (x) ≤ 1 and ϕ(x)r < 1, and so " m m X X ϕ(x)[T f (x) − f (x)] =ϕ(x) hi (x)Sf (yi ) − hi (x))f (x) i=1

i=1

+

m X

i=1

hi (x)f (x) − f (x)

i=1

=

X

hi (x)ϕ(x)[S(f (yi )) − f (x)]

i∈Ix

+ ∈

" X

X i∈Ix

⊆

m X

#

i∈Ix

#

hi (x) − 1 ϕ(x)f (x)

X hi (x) − 1]ϕ(x)rW2 hi (x)W1 + [ i∈Ix

hi (x)W1 + W2

i=1

⊆{W1 + ... + W1 (n + 1)-terms} + W2 ⊆ W. Case II. Suppose x ∈ X\∪m i=1 Hi . Then each hi (x) = 0 and ϕ(x)r < 1 and so ϕ(x)[T f (x) − f (x)] = ϕ(x)[0 − f (x))] ∈ ϕ(x)rW2 ⊆ W. Thus (1) holds.

Theorem 4.5.2. [Kh95] Suppose every compact subset of X has finite covering dimension and E has the approximation property. Then (C(X, E), k) has the approximation property. Proof. Let A be a k-precompact subset of C(X, E), and let K be a compact subset of X and W ∈ W. Suppose K has covering dimension n, and choose a balanced set V ∈ W such that V + V + ... + V ((n + 1)-terms) ⊆ W. Now choose an open balanced set U ∈ W such that U + U + U + U ⊆ V . Since A is precompact in C(X, E), it is easy to verify that D = ∪{f (K) : f ∈ A} is a precompact subset of E. By hypothesis, there exists a continuous linear map S : E → E of finite rank such that Sa − a ∈ U for all a ∈ D.

5. THE APPROXIMATION PROPERTY

85

Choose an open balanced set U1 ∈ W such that U1 ⊆ U ∩ {a ∈ E : Sa ∈ U}. Since A is precompact, there exists a finite subset A1 of A such that A ⊆ A1 + N(K, U1 ). For each x ∈ K, the set G(x) = ∩ {y ∈ X : g(y) − g(x) ∈ U1 } is an g∈A

open neighborhood of x in X. Then there exists a finite open cover U = {G(x1 ), ..., (G(xp )} (say) of K. By hypothesis, we can choose an open cover V = {Hi , ..., Hm } of K which is a refinement of U and such that each point of K belongs to at most n + 1 members of V. For each Hi , choose an index ji such that Hi ⊆ G(yi ), yi = xji . Put Hm+1 = X\K. Then there exist ϕ1 , ..., ϕm+1 ∈ Cb (X) with 0 ≤ ϕi ≤ 1, ϕi = 0 on X\Hi ,

m+! X

ϕi (x) = 1 for x ∈ K,

i=1

and

Pm+1 i=1

ϕi (x) ≤ 1 for x ∈ X. Define T : C(X, E) → C(X, E) by Tf =

m+! X

ϕi ⊗ Sf (y1), f ∈ C(X, E).

i=1

It is easy to verify that T is a continuous linear map of finite rank. To complete the proof, we show that T f − f ∈ N(K, W ) for all f ∈ A. [Let f ∈ A, and let x ∈ K. Choose g ∈ A1 such that (g − f ) ⊆ U1 . Note that Ix = {i : x ∈ Hi } contains at most n + 1 indices. Now T f (x) − f (x) =

m+! X

ϕi (x)[Sf (yi ) − f (x)] =

i=1

For each i ∈ Ix ,

X

ϕi (x)[Sf (yi ) − f (x)].

i∈Ix

Sf (yi ) − f (x) = S[f (yi ) − g(yi )] + [Sg(yi ) − g(yi )] +g(yi ) − g(x)] + [g(x) − f (x)] ∈ SU1 + U1 + U1 ⊆ V. Hence T f (x) − f (x) ∈

X i∈Ix

ϕi (x)V ⊆ W.]

86


Remark. We mention, as before, that if E is a locally convex TVS, then with slight modification of proofs, Theorems 4.5.1 and 4.5.2 hold without the hypothesis of finite covering dimension.


87

6. Notes and Comments Section 4.1. The Stone-Weierstrass theorem was first proved by M.H. Stone [St37] in 1937 for the space (Cb (X), u) as a generalization of the classical Weierstrass’s approximation theorem of 1885 (see also [St48] for its refinement). Several authors have later generalized it or given its new proof; see, e.g. Hewitt [Hew47], Kaplansky ([Kap51], §3), de Branges [Bra59], Bishop [Bis61], Jewitt [Je63], Machado [Mac77], Brosowski and Deutsch [BD81], and Ransford [Ran84]. The Stone-Weierstrass theorems for (Cb (X), β) and (Cb (X, E), β) were first established by Buck [Buc58] in 1958 for E finite dimensional. Its extension to (Cb (X, E), β), E a locally convex TVS, was later given by Todd [Tod65] and Wells [Wel65], independently. For E any TVS, a special case of the theorem (that Cb (X) ⊗ E is u-dense in Cb (X, E)) was established by Shuchat [Shu72a] for (Cb (X, E), u), assuming X a compact space of finite covering dimension (see also Waelbroeck [Wae73], §. 8.1)). More general forms of the theorem were later obtained by Khan [Kh79, Kh80, Kh95], Katsaras [Kat81], Abel [Ab87] and Prolla [Pro93a, Pro93b] for (Cb (X, E), β), (Cb (X, E), u) and (C(X, E), k), assuming the hypothesis of finite covering dimension on X. The present section contains results of these latter authors. Section 4.2. The ‘weighted approximation problem’, formulated by Nachbin [Nac65] and further studied by Summers [SumW71] for E a real or complex field, consists in finding a characterization of the ωV closure of A in CVo (X, E), where A ⊆ CVo (X, E) an M-submodule with M ⊆ C(X) a Cb (X)-submodule. Prolla [Pro71a, Pro71b, Pro72] extended some results of above authors to the case of E a locally convex TVS. This section contains some results from [Kh85, Kh87] when E is not necessarily locally convex. In particular, these results give extension of ([Nac67], Propositions 1 and 2, p.64) and ([Nac67], Proposition 5, p.66) to E a locally bounded TVS. Section 4.3. The main result is an infinite dimensional version of the Weierstrass polynomial approximation theorem, due to Bruno [Bru84], for P (E, F ), the set of all continuous polynomials of finite type from a TVS E into a TVS F . Earlier results in this direction were obtained by Prenter [Pre70] for E = F real separable Hilbert space and by MachadoProlla [MP72] for E and F real locally convex spaces. The hypotheses on E and F involve the existence of sequences of projections having the Grothendieck approximation property (GAP) (cf. [Scha71], pp.108-115).

88


Section 4.4. C. Todd [Tod65] has used the Stone-Weierstrass theorem to characterize the maximal β-closed Cb (X)-submodules of Cb (X, E) for E a locally convex TVS. In this section, we present the extension of these characterizations to a non-locally convex setting, as given in [Kh84]. These require the additional assumptions on E to have the separation property (or, equivalently, the Hahn-Banach extension property) or the weaker separation property considered by Gregory and Shapiro [GrSh70]. Section 4.5. The approximation property for locally convex spaces was defined by Grothendieck in 1955, who also raised “the approximation problem” (i.e. the problem whether every locally convex TVS E has the approximation property). This problem remained unresolved until 1973 when Enflo [Enf73] gave an example of a separable Banach space without the approximation property. However, despite this development, there still exist some concrete Banach spaces about which it is not yet known whether they have the approximation property (see Singer ([Sin70], Vol.III). As regards the function spaces, Grothendieck [Gro55] was the first to show that if X is a compact Hausdorff space, then (C(X), k·k) has the approximation property. In the case of the space (Cb (X), β), this result was proved by Collins and Dorroh [CD68] for X locally compact and by Fremlin, Garling and Haydon [FGH72] for X completely regular. Bierstedt [Bie75] obtained a similar result for the weighted space (CVb (X), ωV ). In the case of vector-valued functions, Fontenot [Fon74] has shown that if X is completely regular and E a normed space with the metric approximation property, then (Cb (X, E), β) has the approximation property. Further extensions of this result have been obtained by Prolla [Pro77] for E a locally convex TVS and by Katsaras [Kat81] for E any TVS but assuming X to have a finite covering dimension. This section contains results from [Kat81, Kh95].

CHAPTER 5

Maximal Ideal Spaces In this chapter we give characterization of maximal closed ideals in the algebras (Cb (X, A), β) and (C(X, A), k) and also of the maximal ideal spaces of these algebras.

89

90

5. MAXIMAL IDEAL SPACES

1. Maximal Ideals in (Cb (X, A), β) We first show that, if A is topological algebra with jointly continuous multiplication, then (Cb (X, A), β) and (C(X, A), k) become topological algebras with jointly continuous multiplication, as follows. Theorem 5.1.1. Let X be a topological space and A a topological algebra with jointly continuous multiplication. Then (Cb (X, A), β) and (C(X, A), k) are topological algebras with jointly continuous multiplication. Proof. Let W be a base of τ neighborhoods of 0 in A consisting of absorbing and balanced sets. Let f, g ∈ Cb (X, A), and let {fα } and {gα } β

β

be nets in Cb (X, A) such that fα → f and gα → g. We need to show β

that fα gα → f g. Let ϕ ∈ Bo (X), 0 ≤ ϕ ≤ 1, and W ∈ W. Choose a V ∈ W such that V + V + V ⊆ W . Since A is a topological algebra with jointly continuous p multiplication, there exists a U ∈ W such that 2 ′ U ⊆ V. Define ϕ (x) = ϕ(x), x ∈ X. Then ϕ′ ∈ Bo (X). Choose r ≥ 1 such that (ϕ′ f )(X) ⊆ rU, (ϕ′ g)(X) ⊆ rU.

(1)

There exists an index αo such that 1 fα − f, gα − g ∈ N(ϕ′ , U) for all α ≥ αo . r

(2)

Now, for any x ∈ X and α ≥ αo , using (1) and (2), ϕ(x)[fα (x)gα (x) − f (x)g(x)] = ϕ′ (x)[fα (x) − f (x)].ϕ′ (x)[gα (x) − g(x)] +ϕ′ (x)f (x).ϕ′ (x)[gα (x) − g(x)] +ϕ′ (x)[fα (x) − f (x)].ϕ′ (x)g(x) 1 1 ∈ U.U + rU. U + U.rU r r ⊆ V + V + V ⊆ W; that is, fα gα − f g ∈ N(ϕ, W ) for all α ≥ αo . Hence the multiplication is jointly continuous in (Cb (X, A), β). Replacing both ϕ, ϕ′ by χK , K any compact set in X, in the above proof, we conclude that (C(X, A), k) is also topological algebras. We shall require the following version of the Stone-Weierstrass Theorem from Chapter 4 for the characterization of β-closed and k-closed ideals.

1. MAXIMAL IDEALS IN (Cb (X, A), β)

91

Theorem 5.1.2. Let X be a completely regular space of finite covering dimension and E a TVS, and let A be a Cb (X)-(resp. C(X))submodule of Cb (X, E) (resp. C(X, E)). Then the following conditions are equivalent: (1) f belongs to the β-closure (resp. k-closure) of A. (2) For each x ∈ X, f (x) ∈ A(x). In the remainder of this section, we shall assume that A is a topological algebra with jointly continuous multiplication and identity e. Further, all ideals considered are the proper ideals. Theorem 5.1.3. [Kh86] For any x ∈ X and any closed left (resp. right, two-sided) ideal M of A, S(x, M) = {f ∈ Cb (X, A) : f (x) ∈ M} is β-closed left (resp. right, two-sided) ideal in Cb (X, A); if M is maximal, so is S(x, M). Proof. The first part is straightforward. Let M be a maximal closed left ideal in A and x ∈ X, and let J be a left ideal in Cb (X, A) such that S(x, M) ⊆ J. Suppose there is a function g ∈ J such that g(x) ∈ / M. Then g(x) has a left inverse c (say) in E. Since J is a left ideal, (1⊗c)g ∈ J, where 1 ∈ Cb (X) is the function identically 1 on X. But, [(1 ⊗ c)(1 ⊗ g(x))](y) = cg(x) = (1 ⊗ e)(y) for all y ∈ X, and so the identity function 1 ⊗ e ∈ J. This contradiction shows that S(x, M) = J, as required. The following theorem gives us a characterization of β-closed (left, right) ideals in Cb (X, A). Theorem 5.1.4. [Kh86] Suppose X has finite covering dimension. Then every β-closed left (resp. right) ideal J in Cb (X, A) is of the form J = ∩ S(x, Jx ), where, for each x ∈ X, Jx represents a closed left x∈X

(resp. right) ideal in A. Proof. Let J be a β-closed left ideal in Cb (X, A). It is easily seen that, for each x ∈ X, Jx = J(x) is a closed left ideal in A. Now, let f ∈ J and ϕ ∈ Cb (X). Then, for any y ∈ X, ϕ(y)f (y) = ϕ(y)[ef (y)] = (ϕ ⊗ e)(y). Since (ϕ ⊗ e)f ∈ J, we have ϕf ∈ J. Hence J is a Cb (X)-submodule of Cb (X, E) and so, by Theorem 6.1.2, J = {f ∈ Cb (X, E) : f (x) ∈ J(x) for all x ∈ X} = ∩ S(x, Jx ). x∈X

We next consider the β-closed two-sided ideals.

92


Theorem 5.1.5. [Kh86] Suppose X has finite covering dimension and A is a simple topological algebra. Then any β-closed two-sided ideal J in Cb (X, A) is of the form JF = {f ∈ Cb (X, A) : f (x) = 0 for all x ∈ F }, where F ⊆ X is a closed subset. Proof. First note that, for any closed subset F ⊆ X, JF is a β-closed two-sided ideal in Cb (X, A). Now, let J be a β-closed two-sided ideal in Cb (X, A). Then the set F = {x ∈ X : f (x) = 0 for all f ∈ J} is closed in X, and clearly J ⊆ JF . Let g ∈ JF , and suppose that g ∈ / J. Since J is β-closed, by Theorem 5.1.2, there exists a y ∈ X such that g(y) ∈ / Jy = J(y). Therefore, Jy 6= A and so, by hypothesis, Jy = {0}. Hence y ∈ F . But g(y) ∈ / Jy implies that g(y) 6= 0, a contradiction. Thus J = JF , as required. Notice that, in the notation of Theorem 5.1.4, we can write the above J as J = ∩ S(x, {0}). x∈F

We now characterize the maximal β-closed ideals in Cb (X, A). Theorem 5.1.6. [Kh86] Suppose X has finite covering dimension and A is a Q-algebra. Then any maximal β-closed left (resp. right, twosided) ideal in Cb (X, A) is of the form S(x, M), where x ∈ X and M is a maximal left (resp. right, two-sided) ideal in A. Proof. Let J be a maximal β-closed left ideal in Cb (X, A), and let f ∈ Cb (X, A) with f ∈ / J. By Theorem 5.1.2, there exists an x ∈ X such that f (x) ∈ / Jx = J(x). Since A is a Q-algebra, there exists a maximal (closed) proper left ideal M in A such that Jx ⊆ M. Clearly, J ⊆ S(x, M), and so, by the maximality of J and Theorem 5.1.3, J = S(x, M). We next state, without proof, the analogues of Theorems 5.1.3-5.1.6 for the algebra (C(X, A), k). Lemma 5.1.7. For any x ∈ X and any closed left (resp. right, two-sided) ideal M of A, S(x, M) = {f ∈ C(X, A) : f (x) ∈ M} is k-closed left (resp. right, two-sided) ideal in C(X, A); if M is maximal, so is S(x, M). The following theorem gives us a characterization of k-closed (left, right) ideals in C(X, A). Theorem 5.1.8. [Kh86] Suppose X has finite covering dimension. Then every k-closed left (resp. right) ideal I in C(X, A) is of the form

1. MAXIMAL IDEALS IN (Cb (X, A), β)

93

I = ∩ S(x, Ix ), where, for each x ∈ X, Ix represents a closed left (resp. x∈X

right) ideal in A. Theorem 5.1.9. [Kh86] Suppose X has finite covering dimension and A is a simple topological algebra. Then any k-closed two-sided ideal J in C(X, A) is of the form J = JF = {f ∈ C(X, A) : f (x) = 0 for all x ∈ F }, where F ⊆ X is a closed subset. Notice that, in the notation of Theorem 5.1.8, we can write the above I as I = ∩ S(x, {0}). x∈F

Theorem 5.1.10. [Kh86] Suppose X has finite covering dimension and A is a Q-algebra. Then any maximal k-closed left (resp. right, twosided) ideal in C(X, A) is of the form S(xo , Mo ), where xo ∈ X and Mo is a maximal left (resp. right, two-sided) ideal in A. Remarks. (1) In general, a maximal ideal in Cb (X, A) need not be β-closed. For instance, if A admits a discontinuous multiplicative linear functional ϕ and M = ϕ−1 (0), then, for any x ∈ X, S(x, M) is a maximal ideal but not β-closed. Thus, if A admits no discontinuous multiplicative linear functional, then every maximal ideal in Cb (X, A) is β-closed. This remark also applies to (C(X, A), k). (2) If the algebra A is locally convex, then Theorems 5.1.4 - 5.1.6 and 5.1.8 - 5.1.10 hold without restricting X to have finite covering dimension.

94


2. Maximal Ideal Space of (Cb (X, A), β) Let X be a completely regular space and A be a topological algebra with jointly continuous multiplication over C and with non-trivial ∗ dual A Recall that ∆(A) (resp. ∆c (A)) denotes the set of all nonzero multiplicative (multiplicative continuous) linear functionals on A. The set ∆c (A) equipped with the Gelfand topology, (i.e., the relative ∗ ∗ ∗ w = w(A , A)-topology induced from A ) is called the maximal ideal space (or the carrier space) of A. We shall assume that ∆c (A) is nonempty. In the literature, ∆c (A) is also called the space of continuous homomorphisms of A, or equivalently the space of closed maximal ideals with codimension 1 of A. For each x ∈ A, let x b : ∆c (A) → C be the Gelfand function defined by x b(ϕ) = ϕ(x) for each ϕ ∈ ∆c (A) and let b = {b A x : x ∈ A}. The mapping x → x b, for each x ∈ A, will be called the Gelfand transform. It is clear that the Gelfand transform is an algebra homomorphism from A into C(∆c (A)). It is well-known (see e.g. [Simm62]) that if X is a compact (resp. completely regular) Hausdorff space, then the maximal ideals in Cb (X) are in one-one correspondence with the points in X (resp. βX); i.e., ∆c (Cb (X)) ∼ = X (resp. ∆c (C(X)) ∼ = βX). For the study of the ideal structures of vector-valued algebras (Cb (X, A), β) and (C(X, A), k), the situation is not so simple. However, the space ∆c (C(X, A)) has been studied in many papers under various kinds of topological assumptions on X, A and C(X, A). Further, results similar in spirit to ∆c (C(X, A)) ∼ = X × ∆c (A) have also been obtained by various authors. In this section, we present results concerning ∆c ((Cb (X, A), β) ∼ = X× ∆c (A). We follow the approach of Prolla [Pro81], but with no convexity assumptions is made about A. We begin with the following classical result which gives a characterization of ∆c (C(X), k). Theorem 5.2.1. (i) The k-closed ideals of C(X) are of the form IF = {f ∈ C(X) : f |F = 0}, where F is a closed ideal in X. (ii) For any x ∈ X, define δx : C(X) → K by δx (f ) = f (x), f ∈ C(X). Then δx ∈ ∆c (C(X)) and the map G : x → δx is a homeomorphism. Thus X ∼ = ∆c ((C(X), k)). Proof. ([Diet69], p. 203) If I is an ideal in C(X), let F = ∩{z(f ) : f ∈ I} = {x ∈ X : f (x) = 0 for all f ∈ I}.

2. MAXIMAL IDEAL SPACE OF (Cb (X, A), β)

95

Then F is closed in X and I is contained in the closed ideal IF . To show that IF ⊆ I, it is sufficient to show thatI is k-dense in IF . [Let h ∈ IF , K ⊆ X a compact set and ε > 0. Then I|K = {f |K : f ∈ I} is a subalgebra of C(X). It is actually an ideal, since if g ∈ C(K), g extends to a g1 ∈ C(X) and g(f |K) = g1 f |K ∈ I|K . Using the Tietze extension theorem and the Urysohn lemma, as in ([Simm62], pp.329-330), I|K is norm dense in {f ∈ C(K) : f |F ∩ K = 0}, since F ∩ K = ∩{z(f |K) : f ∈ I}. Since h|F ∩ K = 0, there is a g ∈ I with pK (g − h) ≤ ε. Thus h belongs to the k-closure of I; that is, k-cl(I) = IF .] So, if I is k-closed, I = IF . (ii) Clearly δx : C(X) → K is a non-zero continuous multiplicative linear functional. We first show that the map G : x → δx is one-one. [Let x 6= y ∈ X. Since X is completely regular, there exist an f ∈ C(X) such that f (x) = 0 and f (y) = 1. Then G(x)(f ) = 0 and G(y)(f ) = 1, and so G(x) 6= G(y).] We next show that G is onto. [Let ϕ ∈ ∆c (C(X)). Since ker ϕ is a closed maximal ideal of C(X), by (i), there is a closed set F such that ker ϕ = IF . Since ker ϕ is maximal (and proper), F reduces to a point x ∈ X. So ker ϕ = ker δx and thus ϕ = δx ([Simm62], p. 321).] So G is a bijection. Finally, G is continuous and open, as follows. [For f ∈ C(X), define fb : ∆c (C(X)) → K by fb(ϕ) = ϕ(f ), ϕ ∈ ∆c (C(X)).

The relative topology w(C(X)∗, C(X)) on ∆c (C(X)) is generated by {fb : f ∈ C(X)}, and so {fb−1 (V ) : V open in K, f ∈ C(X)}

is a subbase for the topology on ∆c (C(X)). Since X is completely regular, its topology equals the weak topology generated by C(X), and so {f −1 (V ) : V open in K, f ∈ C(X)} is a subbase for the topology on X ([Simm62], p. 134). But fbG(x) = fb(δx ) = δx (f ) = f (x), so G−1 (fb−1 (V )) = fbG−1 (V ) = f −1 (V );

that is, G is continuous and open.] Since G is a bijection, it is a homeomorphism. Definition. ∆c (A) is called locally equicontinuous if each h ∈ ∆c (A) has an equicontinuous neighborhood; this always happens in the case of A a normed algebra, since in this case ∆c (A) is a subset of the unit ball of E ∗ .

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Notation. If A is a topological algebra, we can now define a mapping G : X × ∆c (A) → ∆c (C(X, A)) by G(x, ϕ) = ϕ ◦ δx , (x, ϕ) ∈ X × ∆c (A),

(∗)

where δx : C(X, A) → A is the algebra homomorphism given by δx (f ) = f (x), f ∈ C(X, A). Theorem 5.2.2. Let X be a locally compact space of finite covering dimension and A a topological algebra with identity, and let G be the mapping from X × ∆c (A) into ∆c ((Cb (X, A), β),as defined in (∗). Then: (a) G is a bijection. (b) The inverse mapping G−1 is continuous. (c) If ∆c (A) is locally equicontinuous, then G itself is continuous. Proof. Obviously, by (∗), G(x, ϕ) ∈ ∆c (Cb (X, A)) if (x, ϕ) ∈ X × ∆c (A). (a) To show that G is one-one, let (x, h) 6= (y, k) be given in X × ∆c (A). If x = y, then h 6= k. Choose u ∈ A such that h(u) 6= k(u). Since X is locally compact, Cb (X, E) is essential in the sense that we can choose f ∈ Cb (X, E) such that f (x) = u. Then G(x, h)(f ) = h(f (x)) = h(u) 6= k(u) = G(x, k)(f ). If x 6= y, choose ϕ ∈ Cb (X) such that ϕ(x) = 0 and ϕ(y) = 1. Choose now u ∈ A with k(u) = 1. Choose f ∈ Cb (X, E), say f = ϕ ⊗ u, such that f (y) = u. Then g = ϕf ∈ Cb (X, E) and we have G(x, k)(g) = h(g(x)) = h(0) = 0, but G(y, k)(g) = k(g(y)) = k(u) = 1. To show that G is onto, let H ∈ ∆c (Cb (X, A)). Then I = ker H is a proper closed two-sided maximal ideal of (Cb (X, A), β). By Theorem 5.2.1, there exists a unique x ∈ X such that the ideal Ix = cl(I(x)) is proper. Choose fo ∈ Cb (X, A) such that H(fo ) = 1. Then fo is a modular identity for I. For each u ∈ A, uf ∈ Cb (X, A). Define h = hfo : A → K by setting h(u) = hfo (u) := H(ufo ), u ∈ A. ∗

Clearly, h ∈ A . Now, if u, t ∈ A, then uf ∈ I. Choose g ∈ C(X, E) with g(x) = u. Then, since H(fo ) = 1, h(ut) = H(utfo) = H(fo )H(utfo ) = H(fo utfo ) = H(fo u)H(tfo) = H(fou)H(fo )h(t) = H(fo ufo )h(t) = H(fo )H(ufo)h(t) = h(u)h(t);

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hence h is multiplicative. Let J = ker h. We claim that J ⊆ I(x). [Indeed, if u ∈ J, then uf ∈ I. Choose g ∈ Cb (X, A) such that g(x) = u. Then gfo −g ∈ I, and therefore ufo (x)−u ∈ I(x). Now ufo (x) = ufo (x) ∈ I(x), so u ∈ I(x).] Since Ix is proper, it now follows that h 6= 0, i.e., h ∈ ∆c (A). Let L = ker(h ◦ δx ). We claim that I ⊆ L. [Indeed, let g ∈ Cb (X, A) be such that g ∈ / L. Then g(x) ∈ / J. On the other hand, J is a maximal ideal and J ⊆ I(x) ⊆ Ix . Since Ix is proper, it follows that J = Ix . Hence g(x) ∈ / Ix . By Theorem 5.2.1, g ∈ / I.] Since I is maximal and L is closed and proper, I = L. This shows that H and h ◦ δx have the same kernel. Since both are multiplicative, H = h ◦ δx = G(x, h). (c) Using the assumption that ∆c (A) is locally equicontinuous, G is continuous, as follows. Let (xo , ho ) ∈ X × ∆c (A), ε > 0 and g ∈ Cb (X, E) be given. Choose an equicontinuous neighborhood N of ho in ∆c (A) such that N ⊆ {h ∈ ∆c (A) : |(h − ho )(g(xo ))| < ε/2. Let W be a neighborhood of g(x) in A such that |h(w − g(xo ))| < ε/2 for all w ∈ W, h ∈ N. Next, choose a neighborhood U of xo in X such that g(x) ∈ W for all x ∈ U. Then (x, h) ∈ U × N implies g(x) ∈ W and h ∈ N. Therefore |h(g(x) − g(xo ))| < ε/2 and |(h − ho )(g(xo ))| < ε/2. It follows that |G(x, h) − G(xo , ho )(g)| = |h(g(x)) − ho (g(xo ))| ≤ |(h − ho )(g(xo ))| + |h(g(x) − g(xo ))| < ε for all (x, h) ∈ U × N. (b) To show that G−1 is continuous, let Hα → H in ∆c (Cb (X, A)). Since G is onto, there exist nets {xα } in X and {hα } in ∆c (A) such that Hα = G(xα , hα ), and points x in X and h in ∆c (A) such that H = G(x, h). Since H 6= 0, there is some f ∈ Cb (X, A) such that H(f ) = 1. Choose αo such that Hα (f ) 6= 0 for all α ≥ αo . Take any g ∈ Cb (X). Then gf ∈ Cb (X, A) and, for all α ≥ αo , g(xα ) =

g(xα )hα (f (xα )) Hα (gf ) = . hα (f (xα )) Hα (f )

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Now g(xα ) →

H(gf ) H(f ) = H(gf ) = h(g(x))f )x) = g(x)h(f (x)) = g(x)H(f ) = g(x).

Since X is a completely regular Hausdorff space and g ∈ Cb (X) was arbitrary, xα → x. Next, for any u ∈ A, uf ∈ Cb (X, A) and so for α ≥ αo , hα (u)hα (f (xα )) Hα (uf ) hα (u) = = . hα (f (xα )) Hα (f ) Hence H(uf ) hα (u) → = H(uf ) = h(uf (x)) = h(u)h(f (x)) = h(u)H(f ) = h(u), H(f ) i.e., hα → h in the relative weak topology of ∆c (A). Thus G−1 (Hα ) = G−1 [G(xα , hα )] = (xα , hα ) → (x, h) = G−1 (H), and so G−1 is continuous.

Corollary 5.2.3. Let X be a completely regular space of finite covering dimension and A topological algebra with identity, and let G be the mapping from X × ∆c (A) into ∆c ((Cb (X, A), β), as defined in (∗). If ∆c (A) is locally equicontinuous, then G is a homeomorphism. Theorem 5.2.4. Let X be a completely regular space of finite covering dimension and A topological algebra with identity, and let G be the mapping from X × ∆c (A) into ∆c ((C(X, A), k), as defined in (∗). If ∆c (A) is locally equicontinuous, then G is a homeomorphism. Theorem 5.2.5. Let X be a locally compact space of finite covering dimension and A topological algebra with identity, and let G be the mapping from X × ∆c (A) into ∆c ((Co (X, A), u), as defined in (∗). If ∆c (A) is locally equicontinuous, then G is a homeomorphism. Note. If the algebra A is locally convex, then 5.2.2-5.2.5 hold without restricting X to have finite covering dimension.


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3. Notes and Comments Section 5.1. The classical results on characterization of maximal and closed ideals in (C(X, A), k) and (Cb (X, A), u) were obtained by Kaplansky [Kap47] in the case of A a topological ring. These results are further studied in ([Nai72], Ch.V, §26.2) for A a Banach algebra and in ([Pro77], Ch.1, § 6) for A a locally convex algebra. In 1958, Buck [Buc58] showed that there is a one-one correspondence between the maximal β-closed ideals in Cb (X) and the points of X. This section contains characterizations of the maximal and β-closed (resp. k-closed) ideals in Cb (X, A) (resp. C(X, A) where X is a completely regular Hausdorff space of finite covering dimension and A a Hausdorff topological algebra (not necessarily locally convex) with jointly continuous multiplication, as given in [Kh86]. Section 5.2. Concerning the results similar in spirit to ∆c (C(X, A)) ∼ = X × ∆c (A), Hausner ([Hau57], p. 248) showed that if X is compact and A a commutative Banach algebra, then ∆c ((C(X, A), u)) ∼ = X × ∆c (A). This was an improvement of an earlier result of Yood [Y51], Theorem 3.1, Lemma 5.1), where A is a commutative B ∗ algebra with identity (or A a commutative Banach algebra with identity and assuming that every maximal ideal in C(X, E) was of the form S(x, M) = {f ∈ C(X, E) : f (x) ∈ M}, with x ∈ X and M a maximal ideal in A). Using tensor products, Mallios ([Mal66], Theorem 5.1), showing ∆c ((Co (X, A), u)) ∼ = X × ∆c (A), extended the Hausner-Yood theorem to the case of X a locally compact space and A a complete locally m-convex algebra Q-algebra for which ∆c (A) is locally equicontinuous; and (again using tensor products) Dietrich ([Diet69], Theorem 4) showed that ∆c ((C(X, A, k)) ∼ = X × ∆c (A) if X is a completely regular k-space and A is a complete locally convex algebra with ∆c (A) locally equicontinuous. Prolla [Pro81] obtained a similar result for the weighted function algebra (CVo (X, A), ωV ),without using tensor product techniques. Hery [Her76] extended some results of the above authors to the case of A a certain commutative topological algebras, not necessarily locally convex, with identity (see also Arhippainen [Arhi92, Arhi96]. This section contains results about ∆c ((Cb (X, A), β) ∼ = X × ∆c (A), following the approach of Prolla [Pro81] without the convexity assumptions on A.

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Some generalizations of Hery’s results to the case when X is realcompact and A is a metrizable algebra or X is a completely regular Hausdorff space and A is a realcompact metrizable algebra are considered in MR (99k:46089) and MR (2002b:46084) and in the case of topological module-algebras in MR (87c:46056) (here are given conditions when all ideals (not necessarily two-sided) are fixed). Moreover, descriptions of ∆c (Cb (X, A), β), ∆c (Co (X, A), u), ∆c (CVo (X, A), u) and ∆c (Crc (X, A), u) are given in [Ab04] as well and, when A is locally convex, in MR (58#30268) and (for closed (maximal, but not maximal closed) one-sided ideals) in MR (84d:46084). In addition, the description of all closed maximal (not maximal closed) modular left (right and two-sided) ideals in subalgebras of C(X; A; S) (the algebra of A-valued continuous functions f on X for which f (S) is relatively compact in A for each S ∈ S, S is a cover of X which is closed with respect to finite unions) is given in MR (2005c:46064) and in MR (2004a:46045) (the description ∆c (A) of A, where A is a subalgebra of C(X; A; S), is included) by Mart Abel in case when X is a completely regular Hausdorff space, S is a compact cover of X and A is a locally m-pseudoconvex Hausdorff algebra, a locally pseudoconvex Waelbroeck Hausdorff algebra or an exponentially galbed Hausdorff algebra with bounded elements. Similar results for real topological algebras are given by Olga Panova in MR 2326371 and in [O. Panova, Real Gelfand-Mazur algebras. Dissertation, University of Tartu, Tartu, 2006, Dissertationes Mathematicae Univesitatis Tartuensis, 48. Tartu University Press, Tartu, 2006, 84 pp.]. Most recently, Oubbi [Oub07] has given several interesting results related to the above ones.

CHAPTER 6

Separability and Trans-separability In this chapter, we consider various characterizations of separability and trans-separability of the function spaces (Cb (X, E), β), (C(X, E), k), (Cb (X, E), σ), and (Cb (X, E), u).

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1. Separability of Function Spaces Recall that X is called submetrizable if it can be mapped by a one-toone continuous function onto some metric space Z (or, equivalently, if its topology is stronger than a metrizable topology on X). If, in addition, Z is separable, X is called separably submetrizable. A submetrizable space is Hausdorff but not necessarily regular. A topological space X is separably submetrizable iff there exists a sequence {gn } ⊆ Cb (X) which separates points of X. The space ℓ∞ (N) is submetrizable under its weak topology w(ℓ∞ (N), M(N)), where M(N) is the dual of C(N). The space M[0, 1] is a non-separable Banach space, although the weak topology of M[0, 1] is submetrizable. A TVS is submetrizable iff its origin is a Gδ . We state the following three theorems without proof for reference purpose. Theorem 6.1.1. [KK40, [SZ57] (i) (Cb (X), u) is separable iff X is compact metric space. (ii) If X is locally compact, then (Co (X), u) is separable iff X is σ-compact metric space. Theorem 6.1.2. [War58, GuSc72, SumW72] The following statements are equivalent: (a) (Cb (X), β) is separable. (b) (C(X), k) is separable. (c) X is separably submetrizable. Theorem 6.1.3. [GuSc72] The following statements are equivalent: (a) (Cb (X), t) is separable, where t is either σo , σ or u. (b) X is a compact metric space. We now present the extension of above results to the case of E-valued functions. We first obtain an extension of Theorem 7.1.2. Theorem 6.1.4. [Kat81, Kh86b] The following are equivalent: (a) (Cb (X) ⊗ E, β) is separable. (b) (C(X) ⊗ E, k) is separable. (c) X is separably submetrizable and E is separable. Proof. (a) ⇒ (b) If (Cb (X)⊗E, β) is separable, then (Cb (X)⊗E, k) is also separable. Since Cb (X) is k-dense in C(X), it follows that Cb (X)⊗E is k-dense in C(X) ⊗ E. Thus (C(X) ⊗ E, k) separable. (b) ⇒ (c) We first note the fact that both (C(X), k) and E are isomorphic to subspaces of (C(X) ⊗ E, k) via the maps g → g ⊗ a (0 6= a ∈ E fixed) and b → 1X ⊗ b, respectively. Hence, by (b), (C(X), k) is separable and so, by Theorem 6.1.2, X is separably submetrizable. Further, E is also separable.

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(c) ⇒ (a) Suppose X is separably submetrizable. Then, by Theorem 6.1.2, (Cb (X), β) is separable. Let {ϕm } and {an } be countable dense subsets of (Cb (X), β) and E, respectively. Let A be the countable subspace generated by {ϕm ⊗ an : m, n = 1, 2, ...} over rationals. We show that A is β-dense in Cb (X) ⊗ E. Let f ∈ Cb (X) Pp ⊗ E, ϕ ∈ Bo (X) with 0 ≤ ϕ ≤ 1, and W ∈ W. We can write f = i=1 θi ⊗ bi (θi ∈ Cb (X), bi ∈ E). Let V ∈ W be balanced with V + V + ... + V (2p-terms) ⊆ W . Choose r ≥ 1 such that each bi ∈ rV . Let s = max{||θi || : i ≤ i ≤ p}. For each i = 1, ..., p, there exist ϕmi ∈ {ϕm } and ani ∈ {an } such that ||ϕ(ϕmi − θi )|| < 1/r(s + 1)Pand ani − bi ∈ (1/r(s + 1))V . Note that ||ϕϕmi || < s + 1. Let g = ni=1 ϕmi ⊗ ani . Then g ∈ A and, for any x ∈ X, ϕ(x)[g(x) − f (x)] =

p X

ϕ(x)ϕmi (x)[ani − bi ]

i=1

+

p X

ϕ(x)[ϕmi (x) − θi (x)]bi

i=1

1 1 [V + ... + V (p-terms)] + [V + ...V (p-terms)] r s+1 ⊆ W. ∈

Hence g − f ∈ N(ϕ, W ), as required.

Corollary 6.1.5. Let X be a locally compact σ-compact space. Then (Cb (X) ⊗ E, β) is separable iff X is metrizable and E is separable. Corollary 6.1.6. Suppose either X has finite covering dimension or E is locally convex, or E is complete metrizable with a basis. Then the following are equivalent: (a) (Cb (X, E), β) is separable. (b) (C(X, E), k) is separable. (c) X is separably submetrizable and E is separable. In fact, each of the above restrictions on X or E implies that Cb (X)⊗ E is β-dense in Cb (X, E) and that C(X) ⊗ E is k-dense in C(X, E) (see Chapter 5). It is not known whether or not these ‘density’ results hold for E a locally bounded space. However, we can prove Theorem 6.1.7. [Kh86b] Let X be any Hausdorff space and E any locally bounded TVS. Then (Cb (X, E), β) is separable iff (C(X, E), k) is so. Proof. Let {fn } be a countable k-dense subset of C(X, E). Let V be a balanced bounded neighborhood of 0 in E, and let S be a closed

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shrinkable neighborhood of 0 with S ⊆ V . The Minkowski functional ρ = ρS of S is continuous (see Section A.4) and positive homogeneous and, consequently, for each m = 1, 2, ..., the function hm : E → E defined by a if a ∈ mS hm (a) = m a if a ∈ E\mS ρ(a)

is continuous. Further, hm (E) ⊆ mS ⊆ mV, which shows that the functions hm ◦ fn ∈ Cb (X, E). We show that {hm ◦ fn : m, n = 1, 2, ...} is β-dense in Cb (X, E). Let f ∈ Cb (X, E), ϕ ∈ Bo (X) with 0 ≤ ϕ ≤ 1, and W ∈ W. Choose r ≥ 1 such that V + V ⊆ rSV + V ⊆ rW . Choose an integer M ≥ 1 with f (X) ⊆ (M/r)V . Let K be a compact subset of X such that ϕ(x) < 1/rM for x ∈ X\K. There exists an integer N such that (fN − f )(K) ⊆ (1/r)V . Let y ∈ X. If y ∈ K, then fN (y) ∈ MS and so ϕ(y)(hN ◦ fN (y) − f (y)) = ϕ(y)(fN (y) − f (y)) ∈ W. If y ∈ X\K, then ϕ(y)[hM ◦ fN (y) − f (y)] ∈ ϕ(y)[MS − ⊆

M V] r

1 M [V − V ] ⊆ V + V ⊆ W. rM r

Thus hM ◦ fN − f ∈ N(ϕ, W ). Conversely, suppose (Cb (X, E), β) is separable. Then (Cb (X, E), k) is also separable. So we only need to show that Cb (X, E) is k-dense in C(X, E) (cf. Theorem 1.1.11). Let f ∈ C(X, E) and K a compact subset of X. If V is a bounded neighborhood of 0 in E, let S be a closed shrinkable neighborhood of 0 with S ⊆ V . Choose r ≥ 1 with f (K) ⊆ rS. Then, as in the above proof, we get a function hr ◦ f ∈ Cb (X, E) such that hr ◦ f (x) − f (x) = 0 for all x ∈ K. Theorem 6.1.8. [KR91] The following statements are equivalent: (a) (Cb (X) ⊗ E, t) is separable, where t is either σo , σ, or u. (b) X is a compact metric space and E is separable. Proof. (a) ⇒ (b) We first note the fact that both (Cb (X), t) and E are isomorphic to subspaces of (Cb (X) ⊗ E, t) via the maps g → g ⊗ a (0 6= a ∈ E fixed) and b → 1X ⊗ b, respectively. Hence, by (a), (Cb (X), t) is separable and so, by Theorem 6.1.3, X is a compact metric space. Further, E is also separable. (b) ⇒ (a) By Theorem 6.1.1 (i), (Cb (X), u) is separable. Let {gm : m = 1, 2, ...} and {gn : n = 1, 2, ...} be countable dense subsets of (Cb (X), u) and E, respectively and let A consist of finite sums of the

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elements {gm ⊗ an : m, n = 1, 2, ...}. Let f ∈ Cb (X) ⊗ E and W ∈ W any neighborhood of 0 in E. The function f may be written as f = P p i=1 fi ⊗ bi , where fi ∈ Cb (X) and bi ∈ E. Choose a V ∈ W such that V + V + ... + V (2p-terms) ⊆ W and choose r ≥ 1 such that bi ∈ rV (1 ≤ i ≤ p). Let s = max{kfi k : 1 ≤ i ≤ p}. Then, for each 1 ≤ i ≤ p, there exist gmi ∈ {gm } and ani ∈ {an } such that 1 1 and ani − bi ∈ V. kfi − gmi k < r(s + 1) r(s + 1) P Let g = pi=1 gmi ⊗ ani . Then g ∈ A and, for any x ∈ X, g(x) − f (x) =

p X i=1

gmi (x)[ani − bi ] +

p X

gmi (x) − f (x)]bi

i=1

1 1 (V + ... + V (p-terms)) + (V + ... + V (p-terms)) r (s + 1) ⊆ W. ∈

Thus g − f ∈ N(X, W ), and so A is u-dense in Cb (X) ⊗ E.

Corollary 6.1.9. [KR91] Suppose (Cb (X, E), u) is metrizable and Cb (X) ⊗ E is u-dense in Cb (X, E). Then the following statements are equivalent: (a) (Cb (X, E), t) is separable, where t is either σo , σ or u. (b) X is a compact metric space and E is separable. Theorem 6.1.10. (Cb (X, E), u) is separable iff (Cb (X, E), σo ) is separable. Proof. If (Cb (X, E), u) is separable, then clearly (Cb (X, E), σo ) is separable. Suppose that (Cb (X, E), σo ) is separable but that (Cb (X, E), u) is not. Then, for any sequence {fn : n = 1, 2, ...} in Cb (X, E), there exist an f ∈ Cb (X, E), W ∈ W, and a sequence {xn } in X such that fn (xn )−f (xn ) ∈ / W for all n. Thus, if A = {xn }, then fn −f ∈ / N(χA , W ). This contradicts the separability of (Cb (X, E), σo ). Next, using Theorem 6.1.1(ii) instead of Theorem 6.1.3, we can easily obtain: Theorem 6.1.11. Let X be locally compact. Then (Co (X) ⊗ E, u) is separable iff X is a σ-compact metric space and E is separable. Corollary 6.1.12. [KR91] Suppose that either X has finite covering dimension or E locally convex, or E has the approximation property. Then (i) (Cb (X, E), t) is separable iff X is compact metric space and E is separable, where t is either σ, σo or u.

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(ii) If X is locally compact, (Cb (X, E), u) is separable iff X is σcompact metric space and E is separable. Proof. Each of the above restrictions on X or E implies that, in case (i), Cb (X) ⊗ E is u-dense in Cb (X, E) and that, in case (ii), Co (X) ⊗ E is u-dense in Cb (X, E). The result now follows by applying Theorems 6.1.8 and 6.1.11. Finally we give some examples concerning the above results. Examples 6.1.13. For 0 < p < 1, let ℓp = ℓp (N) denote the usual space of scalar sequences, and let hp = hp (D) denote the Hardy space of certain harmonic functions on the unit disc D of the complex plane (see [Shap85]). ℓp and hp are not locally convex but are locally bounded (hence metrizable) and their duals separate points; ℓp is separable while hp is not since it contains a copy of ℓ∞ ([Shap85], Theorem 3.5). Let ω be the first uncountable ordinal and α < ω a limit ordinal. Let Y = [0, ω], Yo = [0, ω), Z = [0, α], and Zo = [0, α), each endowed with the order topology ([Will70, [StSe78). Then 1. Cb (X) ⊗ ℓp is ( a) β-dense in Cb (X, ℓp ) (b) k-dense in C(X, ℓp ), and (c) u-dense (hence also σ-, σo -dense) in Crc (X, ℓp ) for any completely regular Hausdorff space X since ℓp is admissible (see Chapter 4). 2. (Cb (Z, ℓp ), u) is separable since Z is compact and metrizable, but (Cb (Y, ℓp ), u) and (Cb (Yo , ℓp ), u) are not separable since Y and Yo are not metrizable. 3. (Co (Zo , ℓp ), u) is separable since Zo is σ-compact and metrizable; (Co (Yo, ℓp ), u) is not separable since Yo is not σ-compact and also not metrizable. 4. (Cb (Zo , E), β) and (C(Zo , E), k) are separable for E = ℓp but not for E = hp , since Zo is separable and metrizable, ℓp is separable, and hp is not separable.

2. TRANS-SEPARABILITY OF FUNCTION SPACES

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2. Trans-separability of Function Spaces In this section we present some results on trans-separability, as given in [Kh99, Kh04, Kh08b]. Recall that a locally convex TVS L is called seminorm-separable if, for each continuous seminorm p on L, (L, p) is separable. The following two classical results are stated for reference purpose. Theorem 6.2.1. [GuSc72] The following statements are equivalent: (a) (Cb (X), β) is seminorm-separable. (b) (C(X), κ) is seminorm-separable. (c) Every compact subset of X is metrizable. Theorem 6.2.2. [GuSc72] The following statements are equivalent: (a) (Cb (X), σ) is seminorm-separable. (b) (Cb (X), σo ) is seminorm-separable. (c) The closure in βX of each σ-compact subset of X is metrizable in βX. Definition. A uniform space L is called trans-separable if every uniform cover of L admits a countable subcover. In particular, a TVS L is trans-separable if, for each neighborhood W of 0 in L, the open cover {a + W : a ∈ L} of L admits a countable subcover. For TVSs, a generalized notion of separability, namely, the neighborhoodseparability may be defined, as follows. Definition. Let L be a TVS, and let V be a neighborhood of 0 in L. A subset H of L is said to be V -dense in L if, for any z ∈ L and δ > 0, there exists an element y ∈ H such that y − z ∈ δV . L is called neighborhood-separable if, for each neighborhood V of 0, there exists a countable V -dense subset of L. Another notion of generalized separability may also be considered, as follows. Definition. Let (L, τ ) be a TVS whose topology is generated by a family Q(τ ) of continuous F-seminorms. Then (L, τ ) is called Fseminorm-separable if (L, q) is separable for each q ∈ Q(τ ). Clearly, separability implies F-seminorm-separability; the converse holds in metrizable spaces. The following result establishes the equivalence of all the above notions of generalized separability. Theorem 6.2.3. [Kh04, Kh08b] For a TVS (L, τ ), the following are equivalent:

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(1) (L, τ ) is trans-separable. (2) (L, τ ) is neighborhood-separable. (3) (L, τ ) is F-seminorm-separable. Proof. (1) ⇒ (2) Suppose L is trans-separable, and let V be a neighborhood of 0. For each n ≥ 1, Un = {x + n1 V : x ∈ L} is a uniform cover (n) of L, and so it has a countable subcover U∗n = {xk + n1 V : k ∈ N}. Let (n) D = ∪∞ : k ∈ N}. To show that D is V -dense in L, let y ∈ L n=1 {xk and δ > 0. Choose N ≥ 1 such that N1 < δ. Since U∗N is a cover of (N ) (N ) L, y ∈ xK + N1 V for some K ∈ N. Then y − xK ∈ δV . Hence L is neighborhood-separable. (2) ⇒ (1) Suppose L is neighborhood-separable, and let U be a neighborhood of 0. Choose a balanced neighborhood V of 0 with V + V ⊆ U. Let {zn } be a countable V -dense subset of L. Since L = ∪x∈L (x + V ), to each zn ∈ L, there exists some xn ∈ L such that zn − xn ∈ V . Let y ∈ L. Choose zk such that y − zk ∈ V. Then y − xk = (y − zk ) + (zk − xk ) ∈ V + V ⊆ U. Hence L = ∪n≥1 (xn + U), and so L is trans-separable. (1) ⇒ (3) Suppose L is trans-separable, and let q ∈ Q(τ ). For each n ≥ 1, let Vn = {x ∈ L : q(x) < 1/n}, a balanced neighborhood of 0 in L. Then, for each n ≥ 1, Un = {x + Vn : x ∈ L} is a uniform cover of (n) L, and so it has a countable subcover U∗n = {xk + Vn : k ∈ N}. Let (n) D = ∪∞ : k ∈ N}. To show that D is dense in (L, q), let y ∈ L n=1 {xk and ε > 0. Choose N ≥ 1 such that 1/N < ε. Since U∗N is a cover of (N ) (N ) L, y ∈ xK + VN for some K ∈ N. Then q(y − xK ) < 1/N < ε. Hence (L, q) is separable. (3) ⇒ (1) Suppose (L, q) is separable for each q ∈ Q(τ ). Let {x + U : x ∈ L} be any uniform cover of L, where U is neighborhood of 0 in L. Choose a balanced neighborhood V of 0 in L with V + V ⊆ U. Choose q ∈ Q(τ ) such that W = {x ∈ L : q(x) < 1} ⊆ V . Let {zn } be a countable dense subset in (L, q). Since L = ∪x∈L (x + W ), to each zn ∈ L, there exists some xn ∈ L such that zn − xn ∈ W . Let y ∈ L. Choose zk such that q(y − zk ) < 1. Then y − xk = (y − zk ) + (zk − xk ) ∈ W + W ⊆ U, and so L = ∪n≥1 (xn + U).

Theorem 6.2.4. [Kh08b] Let E be any non-trivial TVS. Then the following statements are equivalent: (a) (Cb (X) ⊗ E, β) is trans-separable.


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(b) (C(X) ⊗ E, k) is trans-separable. (c) Every compact subset of X is metrizable and E is trans-separable. Proof. (a) ⇒ (b) This follows from the fact that k 6 β on Cb (X) ⊗ E and that Cb (X) ⊗ E is k-dense in C(X) ⊗ E. (b) ⇒ (c) This follows from Theorem 6.2.1 and the fact that both (C(X), k) and E are isomorphic to subspaces of (C(X) ⊗ E, k) via the maps g → g ⊗ a (0 6= a ∈ E fixed) and a → 1X ⊗ a, respectively. (c) ⇒ (a) By Theorem 6.2.1, (Cb (X), β) is trans-separable. Fix a ϕ ∈ Bo (X), 0 ≤ ϕ ≤ 1 and a balanced W ∈ W. We need to show that there is a countable set H ⊆ Cb (X) ⊗ E such that Cb (X) ⊗ E = H + N(ϕ, W ). For every pair m, n ∈ N choose a balanced Um,n ∈ W so that, denoting Vm,n = Um,n + mUm,n + Um,n , one has Vm,n + · · · + Vm,n (n-summands) ⊆ W. Also, choose a countable set Dm,n in E so that E = Dm,n + Um,n . Let D be the union of all these sets Dm,n (m, n ∈ N). Next, for each k ∈ N denote Bk = {f ∈ Cb (X) : kf kϕ ≤ 1/k} and choose a countable set Gk in Cb (X) so that Cb (X) = Gk + Bk . Let G be the union of all these sets Gk (k ∈ N). We are going to show that theP countable set H = Hϕ,W of all functions in Cb (X) ⊗ E of the form h = ri=1 gi ⊗ di , where gi ∈ G and di ∈ D (i = 1, . . . , r, r ∈ N), is as required. P Take any f ∈ Cb (X)⊗E. Then f = ni=1 fi ⊗ai for some f1 , . . . , fn ∈ Cb (X) and a1 , . . . , an ∈ E. Let m ∈ N be such that kfi kϕ ≤ m for i = 1, . . . , n, and next choose k ∈ N so that k −1 ai ∈ Um,n for i = 1, . . . , n. By the definitions of Dm,n and Gk , there are d1 , . . . , dn ∈ Dm,n and g1 , . . . , gk ∈ Gk such that ai − di ∈ Um,n

and kfi − gi kϕ ≤ 1/k

for i = 1, . . . , n.

Now, for i = 1, . . . , n and x ∈ A, ϕ(x)[fi (x)ai − gi (x)di ] =ϕ(x)[fi (x) − gi (x)]ai + ϕ(x)fi (x)(ai − di ) + ϕ(x)[gi (x) − fi (x)](ai − di );

(*)

hence (using the fact that Um,n is balanced) ϕ(x)[fi (x)ai − gi (x)di ] ∈ Um,n + mUm,n + Um,n = Vm,n . P In consequence, setting h = ni=1 gi ⊗ di we have h ∈ H and for every x ∈ A, ϕ(x)[f (x) − h(x)] =

n X i=1

ϕ(x)[fi (x)ai − gi (x)di ] ∈ W

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so that f − h ∈ N(ϕ, W ).

Remark. A somewhat more transparent variant of the above proof that (c) implies (a) can be based on Theorem 6.2.3. Thus, one has to show that for any ϕ ∈ Bo (X), 0 ≤ ϕ ≤ 1, and any continuous F -seminorm q on E, the space (Cb (X) ⊗ E, pϕ ) is separable, where pϕ (f ) = sup q(ϕ(x)f (x)), f ∈ Cb (X, E) x∈X

Now, let G be a countable subset dense in (C k·kϕ ), and D a countPb (X), n able set dense in (E, q). Take any f = i=1 fi ⊗ ai in Cb (X) ⊗ E, and P choose m ∈ N so that kfi kϕ ≤ m for each i. Given ε > 0, let g = ni=1 gi ⊗ di , where gi ∈ G and di ∈ D. Assume that kfi − gi kϕ ≤ δ for all i and some as yet unspecified 0 < δ < 1. Then, making use of (∗), it is easily seen that pϕ (f − g) ≤

n X

[q(kfi − gi kϕ ai ) + q(kfi kϕ (ai − di )

i=1

+q(kfi − gi kϕ (ai − di ))] n X ≤ [q(δai ) + (m + 1)q(ai − di )] i=1

and this can be made smaller than ε by taking δ sufficiently small and choosing the di ’s in D sufficiently close to the ai ’s. It follows that the countable set of all g’s of the above form is dense in (Cb (X) ⊗ E, pϕ ). Next, we obtain: Theorem 6.2.5. [Kh04] Let E be a non-trivial TVS. Then: (a) (Cb (X) ⊗ E, u) is trans-separable iff X is a compact metric space and E is trans-separable. (b) Suppose X is locally compact. Then (Co (X) ⊗ E, u) is transseparable iff X is a σ-compact metric space and E is trans-separable. Proof. (a) In this case, (Cb (X) ⊗ E, u) is trans-separable iff it is separable iff X is a compact metric space. The proof now follows just as in above theorem. (b) If X is locally compact, then (Co (X), u) is trans-separable iff it is separable iff X is a σ-compact metric space. Again the proof follows just as in the above theorem. Again we remark that, if Cb (X) ⊗ E (resp. Co (X) ⊗ E) is u-dense in Cb (X, E) (resp. Co (X, E)), the above theorem remains valid with Cb (X) ⊗ E (resp. Co (X) ⊗ E) replaced by Cb (X, E) (resp. Co (X, E)).


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Remark.If X has finite covering dimension or E is locally convex, or E has the approximation property or E is complete metrizable with a basis, then Cb (X) ⊗ E is β-dense in Cb (X, E) and that C(X) ⊗ E is k-dense in C(X, E) (see Chapter 4). Hence, under these assumptions, the above two theorems hold with Cb (X) ⊗ E, C(X) ⊗ E and Co (X) ⊗ E replaced by Cb (X, E), C(X, E) and Co (X, E), respectively. It is not known whether or not these ‘density’ results hold for E a locally bounded space. However, we can prove Theorem 6.2.6. [Kh08b] Let X be any Hausdorff space and E any locally bounded space. Then (Cb (X, E), β) is trans-separable iff (C(X, E), k) is so. Proof Suppose (C(X, E), k) is trans-separable. Let ϕ ∈ Bo (X) with 0 ≤ ϕ ≤ 1, and let W ∈ W. Let V be a balanced bounded neighborhood of 0 in E, and let S be a closed shrinkable neighborhood of 0 with S ⊆ V . The Minkowski functional ρ = ρS of S is continuous and positive homogeneous and, consequently, for each r > 0, the function hr : E → E defined by a if a ∈ rS hr (a) = r a if a ∈ E \ rS ρ(a) is continuous. Further, hr (E) ⊆ rS ⊆ rV, which shows that, for each f ∈ C(X, E), the function hr ◦ f ∈ Cb (X, E). Choose t ≥ 1 such that V +V ⊆ tS and V +V ⊆ tW . For each m = 1, 2, ..., there exists a compact set Km ⊆ X such that ϕ(x) < 1/tm2 for x ∈ X\Km . Corresponding to each Km , choose {fmn : n = 1, 2, ...} as a N(Km , V )-dense of C(X, E). We show that {hm ◦ fmn : m, n = 1, 2, ...} is β-dense in Cb (X, E). Let f ∈ Cb (X, E) and 0 ≤ δ ≤ 1. Choose integers M ≥ 1/δ and N ≥ 1 such that f (X) ⊆ (Mδ/t)V and (fM N − f )(KM ) ⊆ (δ/t)V . Let y ∈ X. If y ∈ KM , then fM N (y) ∈ f (y) + (δ/t)V ⊆ (Mδ/t)V + (Mδ/t)V ⊆ MS, and so ϕ(y)[hM ◦ fM N (y) − f (y)] = ϕ(y)[fM N (y) − f (y)] ∈ ϕ(y)(δ/t)V ⊆ δW. If y ∈ X\KM , then, since hM (fM N (y)) ∈ hM (E) ⊆ MS, Mδ V] t M δ δ ⊆ [V + V ] ⊆ [V + V ] ⊆ δW. tM 2 t t − f ∈ δN(ϕ, W ).

ϕ(y)[hM ◦ fM N (y) − f (y)] ∈ ϕ(y)[MS −

Thus hM ◦ fM N

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The converse follows from the fact that Cb (X, E) is dense in (C(X, E), k). Indeed, let f ∈ C(X, E), K a compact subset of X and W ∈ W. Let S and V be as above with S ⊆ V . Choose r ≥ 1 with f (K) ⊆ rS. Then, as in the above proof, we get a function hr ◦ f ∈ Cb (X, E) such that hr ◦ f (x) − f (x) = f (x) − f (x) = 0 ∈ W for all x ∈ K.

Similar characterizations of trans-separability for the σ and σo topologies are stated without proof since they can be obtained by applying Theorem 6.2.2 in place of Theorem 6.2.1, with slight modifications. Theorem 6.2.7. [Kh04] Let E be any non-trivial TVS. Then the following statements are equivalent: (a) (Cb (X) ⊗ E, σ) is trans-separable. (b) (Cb (X) ⊗ E, σo ) is trans-separable. (c) The closure in βX of each σ-compact subset of X is metrizable in βX and E is trans-separable. Note If Cb (X) ⊗ E is u-dense in Crc (X, E), then clearly Theorem 7.2.7 holds with Cb (X) ⊗ E replaced by Crc (X, E). We do not know of any general assumption on X other than its compactness ensuring that Cb (X) ⊗ E is σ- or σo -dense in Cb (X, E). However, without this density assumption, the following holds. Theorem 6.2.8. [Kh04] (Cb (X, E), σ) is trans-separable iff (Cb (X, E), σo ) is so. Proof. (⇒) This follows from the fact that σo 6 σ. (⇐) Suppose (Cb (X, E), σ) is not separable. We show that (Cb (X, E), σo ) is also not trans-separable, i.e., if H = {fn } is any countable set in Cb (X, E), then there exist a countable set B = {xn } ⊆ X and a W ∈ W such that Cb (X, E) 6= H + N(B, W ). Since (Cb (X, E), σ) is not separable, there exists a σ-compact set A ⊆ X and a W ∈ W such that Cb (X, E) 6= H + N(B, W ). Then there exists an f ∈ Cb (X, E) such that f − fn ∈ / N(A, W ) for all n ≥ 1. So, for each n ≥ 1, there exists an xn ∈ A such that f (xn ) − fn (xn ) ∈ / W. Then, if B = {xn }, f − fn ∈ / N(B, W ) for all n ≥ 1. Hence Cb (X, E) 6= H + N(B, W ), and so (Cb (X, E), σo ) is not trans-separable, as desired. Next, using Theorem 6.1.1 and the method of proof of Theorem 6.2.5, we obtain: Theorem 6.2.9. [Kh08b] Let E be a TVS. Then:


113

(a) (Cb (X)⊗E, u) is trans-separable iff X is a compact metric space and E is trans-separable. (b) Suppose X is locally compact. Then (Co (X) ⊗ E, u) is transseparable iff X is a σ-compact metric space and E is trans-separable. Again we remark that, if Cb (X) ⊗ E (resp. Co (X) ⊗ E) is u-dense in Cb (X, E) (resp. Co (X, E)), Theorem 6.2.9 remains valid with Cb (X) ⊗ E (resp. Co (X) ⊗ E) replaced by Cb (X, E) (resp. Co (X, E)). Finally, we give some examples concerning the above results and remarks. Examples 6.2.10. Using the notations of Example 6.1.13, we have 1. (Cb (Yo , E), β) and (C(Yo , E), k) are trans-separable for E = ℓp but not for E = hp . Here every compact subset of Yo is metrizable although Yo is not. 2. (Cb (Z, ℓp), u) and (Co (Zo , ℓp ), u) are (trans-) separable (since Z is compact and metrizable and Zo is σ-compact and metrizable) but (Cb (Y, ℓp ), u) and (Co (Yo , ℓp ), u) are not (since Y is compact but not metrizable and Yo is not σ-compact and also not metrizable). 3. (Cb (Y ) ⊗ ℓp , u) and (Cb (Z) ⊗ hp , u) are not trans-separable since Y is not metrizable and hp is not trans-separable. 4. If ωo the first countable infinite ordinal and T = [0, ω] × [0, ωo]\{(ω, ωo)}, the deleted Tychonoff plank, with the product topology [StSe78; Will70], then(Cb (T ) ⊗ ℓp , σ) is not trans-separable since, if A = ∪∞ i=1 [0, ω] × {n}, A is σ-compact and A¯βT = βT = [0, ω] × [0, ωo] which is not metrizable: in fact the point (ω, ωo) is not even a Gδ in T (see [GuSc72], p. 259).

114


3. Notes and Comments Section 6.1. The fundamental result on the characterization of separability of (Cb (X), u) was obtained by M. Krein and S. Krein [KK40] in 1940. Later, similar results were obtained by Warner [War58] for (C(X), k) and by Gulick and Schmets [GuSc72] and, independently, by Summers [SumW72] for (Cb (X), β). The notions of submetrizable and separably submetrizable spaces are given in [War58, SumW72]. In the case of vector-valued functions, Todd [Tod65] and Choo [Cho79] have considered the separability of (Cb (X, E), β) for E a locally convex TVS and by Katsaras [Kat83], Khan [Kh85, Kh86] and Khan-Rowlands [KR91] for E a TVS. Section 6.2. As stated in [RobN91], Trans-separability appears under many different names in the literature. While workers in the field of uniform spaces tend to speak simply of separable space (see [Hag74, Isa64]), those who study locally convex spaces favour the phrase ”seminorm separable”. Pfister [Pf76] investigated TVSs which were of ”countable type”, whereas Drewnowski [Dre75] coined the word ”transseparable” while working with topological abelian groups and this also seems to be suitable in the case of TVSs. This notion has proved to be useful in the work [Pf76, CO87, RobN91] while studying the metrizability of precompact sets in locally convex spaces. Characterizations of seminorm separability for (Cb (X), u), (C(X), k) and (Cb (X), β) were first given by Gulick and Schmets [GuSc72] This section contains results from [Kh99, Kh04, Kh08b] which generalize the results of [GuSc72] to vector-valued functions by considering the notion of neighborhood separability, or equivalently, trans-separability in the non-locally convex setting.

CHAPTER 7

Weak Approximation in Function Spaces In this chapter we are mainly concerned with the weak approximation type results for Cb (X, E) and CVo (X, E). Throughout this chapter we shall use the notations of Chapters 1 and 4 concerning the strict topology β and the weighted topology ωV . As seen in Chapter 4, Cb (X) ⊗ E is β-dense in Cb (X, E) and CVo (X) ⊗ E is ωV -dense in CVo (X, E) for a locally convex E, and also for some concrete classes of not necessarily locally convex spaces E (such as admissible spaces and spaces having the approximation property); the general case of density problem remains open. The main results in this section include a “convexified” version of the approximation problem. It states that, if X is locally compact, then CVo (X)⊗ E is always weakly dense in CVo (X, E). This implies that CVo (X) ⊗ E is dense in CVo (X, E) equipped with the locally convex topology ωVc associated to ωV (i.e., the strongest locally convex topology on CVo (X, E) which is weaker than ωV ), where E c is the locally convex TVS associated to E. These results are used later in Chapter 8 to obtain a representation of β- and ωV -continuous linear functionals on Cb (X, E) and CVo (X, E), respectively.

115

116

7. WEAK APPROXIMATION IN FUNCTION SPACES

1. Weak Approximation in (Cb (X, E), β) In this section, we shall present some results regarding the weak approximation in the space (Cb (X, E), β), as given in [Naw85]. We shall use here the measure theoretic terminology of section A.7. Let k(X), z(X) and cz(X) denote the families of all compact, zero and cozero subsets of X, respectively, let Ba(X) denote the σ-algebra of subsets of X generated by z(X). The family of all positive, tight Baire measures on X will be denoted by Mt+ (Ba(X)). Let C(X, I) denote the subset of Cb (X) of all functions having values in I = [0, 1]. If f is a function on X into E or R and U ∈ cz(X), then f ≺ U means supp(f ) ⊆ Z for some Z ∈ z(X) with Z ⊆ U. We recall that if A ∈ k(X) or A ∈ z(X) and B ∈ cz(X) with A ⊆ B, then: (a) there are sets F ∈ k(X), Z ∈ z(X) such that A ⊆ F ⊆ Z ⊆ B; (b) there is a function θ ∈ C(X, I) such that θ ≺ B and θ(A) = 1. If E is a TVS, then its topology may be generated by some family of F -seminorms (§ A.2). This implies that if W is a base of balanced neighborhoods of 0 in E, then W ∩ Ba(E) is also a base of 0 consisting of balanced and zero or cozero sets. Definition. We define the mt -topology on Cb (X, E) as the linear topology which has a base at 0 the family of all sets of the form G = G(µ, W, ε) = {f ∈ Cb (X, E) : µ({x : f (x) ∈ / W }) ≤ ε},

(∗)

where µ ∈ Mt+ (Ba(X)), W ∈ W ∩ Ba(E), and ε > 0. Let J be the family of all linear topologies τ on Cb (X, E) satisfying τ |H 6 mt |H for any u-bounded subset H of Cb (X, E). We define the γ-topology as supJ . Lemma 7.1.1. [Naw85] If T is a linear functional on Cb (X, E), then the following statements are equivalent: (a) T ∈ (Cb (X, E), γ)∗ . (b) T (fα ) → 0 for every net {fα } ⊆ Cb (X, E) which is u-bounded and mt -convergent to 0. Proof. The implication (a) ⇒ (b) is obvious. (b) implies that the weak topology σ(T ) induced on Cb (X, E) by T belongs to J . Thus σ(T ) 6 γ, and so T is γ-continuous. Lemma 7.1.2. [Naw85] The space Cb (X)⊗ E is γ-dense in Cb (X, E). Proof. Fix f ∈ Cb (X, PE). Let Sf ⊆ Cb (X) ⊗ E be the family of all functions of the form ni=1 θi ⊗ f (xi ), where θi ∈, C(X, I), xi ∈ X, θi (xi ) = 1 and supp(θi ) ∩ supp(θj ) = ∅ if i 6= j, i, j = 1, ..., n, n ∈ N. It is easy to see that Sf is u-bounded. Let G be a mt -neighborhood of 0 of the form G(µ, W, ε) given by (∗). We can assume that W is balanced.

1. WEAK APPROXIMATION IN (Cb (X, E), β)

117

Choose a balanced W1 ∈ W such that W1 + W1 ⊆ W and W1 ∈ cz(E). By the tightness of µ, we can find K ∈ k(X) such that µ(B) ≤ ε/2 for every B ∈ Ba(X), B ∩ K = ∅. The set f (K) is compact, so f (K) ⊆ F + W1 for some finite subset F of E. It follows that there are sets B1 , ..., Bp ∈ Ba(X) such that K ⊆ p

∪ Bi , Bi ∩ Bj = ∅ if i 6= j and

i=1

f (x) − f (y) ∈ W for any x, y ∈ Bi , i = 1, ...., p. Let Zi ∈ z(X), Ui ∈ cz(X) be such that Zi ⊆ Bi ⊆ Ui and µ(Ui \Zi ) ≤ ε/(2p). We can find sets Oi ∈ cz(X) such that Zi ⊆ Oi ⊆ Ui and Oi ∩ Oj = ∅ if i 6= j, i = 1, ..., p. There exist functions θi ∈ C(X, I) such that θi (Zi ) = 1, supp(θi ) = Oi , i = 1, ..., p. Choose xi ∈ Zi , i = 1, ..., p. Then the function h = belongs to Sf and µ({x : (f − h)(x) ∈ / W }) ≤

p X i=1

p

Pp

µ(Ui \Zi ) + µ(X\ ∪ Ui ) ≤ i=1

i=1 θi ⊗

f (xi )

pε + ε/2 = ε. 2p

Therefore f − g ∈ G(µ, W ,ε), so Cb (X) ⊗ E is γ-dense in Cb (X, E).

Theorem 7.1.3. [Naw85] Every β-continuous linear functional on Cb (X, E) is γ-continuous. Proof. Let T ∈ (Cb (X, E), β)∗. There exist ϕ ∈ Bo (X) and W1 ∈ W such that |T (f )| ≤ 1 for all f ∈ N(ϕ, W1 ).

(1)

We may assume that W1 is balanced and belongs to cz(E). Since β 6 u, we can find a balanced W2 ∈ W such that N(1, W2 ) ⊆ N(ϕ, W1 ), W2 ⊆ W1 , W2 ∈ z(E).

(2)

We observe that: For every ε > 0, there is a Kε ∈ K(X) such that |T (f )| ≤ ε for all f ∈ N(1, W2 ) with f (Kε ) = {0}.

(3)

[Indeed, let Kε be a compact subset of X such that {x : |ϕ(x)| ≥ ε} ⊆ Kε . For any f ∈ N(1, W2 ), f (X) ⊆ W1 by (2). If, moreover, f (Kε ) = {0}, then ϕf (X) ⊆ εW1 . Therefore by (1), |T (f )| ≤ ε.]

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We define F : cz(X) → R by F (U) = sup{|T (f )| : f ∈ N(1, W2 ), f ≺ U} for any U ∈ cz(X). Obviously F is positive, finite and if U1 ,U2 ∈ cz(X), U1 ⊆ U2 , then F (U1 ) ≤ F (U2 ). We will show that F (U1 ∪ U2 ) ≤ F (U1 ) + F (U2 ) for any U1 , U2 ∈ cz(X).

(4)

(4) F (U1 ∪ U2 ) ≤ F (U1 ) + F (U2 ) for any U1 ,U2 ∈ cz(X). [Let U1 , U2 ∈ cz(X), f ∈ N(1, W2 ) and f ≺ U1 ∪ U2 . There is a Z ∈ z(X) such that supp(f ) ⊆ Z ⊆ U1 ∪ U2 . Fix ε > 0. Let Kε be as in (3). Put K = Z ∩ Kε , H1 = K ∩ U1 , H2 = K\H1 . For any x ∈ Hi there are sets Ux ∈ cz(X), Zx ∈ z(X) such that x ∈ Ux ⊆ Zx ⊆ Ui , i = 1, 2. The family {Ux : x ∈ K}is an open cover of K. By the compactness of K we can find a finite subset S of K such that K ⊆ ∪{Ux : x ∈ S}. Let Zi′ = ∪{Zx : x ∈ S ∩ Hi }, i = 1, 2. Then Zi′ ∈ z(X), Zi′ ⊆ Ui , i = 1, 2. There are sets Zi ∈ z(X) and Oi ∈ cz(X) such that Zi′ ⊆ Oi ⊆ Zi ⊆ Ui , i = 1, 2. Let θo , θ1 , θ2 ∈ C(X, I) be functions with supports U1 ∪ U2 \(Z1′ ∪ Z2′ ), O1 , O2 respectively. We define functions ϕi = θi (θo + θ1 + θ2 )−1 , i = 0, 1, 2. Let fi = ϕi f , i = 0, 1.2. Then f1 ≺ U1 , f2 ≺ U2 and fo (Kε ) = {0}. Moreover, fi ∈ N(1, W2 ), i = 1, 2, and f = fo + f1 + f2 , so that |T (f )| ≤ |T (fo)| + |T (f1 )| + |T (f2 )| ≤ ε + F (U1 ) + F (U2 ). This implies that F (U1 ∩ U2 ) ≤ F (U1 ) + F (U2 ).] Suppose additionally that U1 ∩ U2 = ∅. Fix ε > 0. Let f1 , f2 ∈ N(1, W2 ) be such that T fi ≥ F (U1 ) + F (U2 ) − ε. Thus F (U1 ∪ U2 ) = F (U1 ) + F (U2 ) for any U1 , U2 ∈ cz(X), U1 ∩ U2 = ∅. (5) From (3) it immediately follows that, for every ε > 0, there exists a Kε ∈ k(X) such that F (U) ≤ εif U ∈ cz(X) and U ∩ Kε = ∅.

(6)

Moreover, for every ε > 0 and U ∈ cz(X), there is a Z ∈ z(X), Z ⊆ U, such that F (U\Z) ≤ ε. (7)

1. WEAK APPROXIMATION IN (Cb (X, E), β)

119

[Indeed, if this statement fails to be true for some ε > 0, then by induction we can find a sequence {fn } ⊆ N(1, W2 ) such that supp(fi ) ∩ supp(fj ) = ∅ for i 6= j and T (fj ) > ε, i, j = 1, 2, .... But gn = f1 +..+ fn belongs to N(1, W2 ) for every n ∈ N and T (gn > nε. This contradicts (1).] We define µ(B) = inf {F (U) : U ∈ cz(X), U ⊇ B} for B ⊆ X. It is easy to see that the family B of all subsets B of X such that for any given ε > 0 there are Z ∈ z(X), U ∈ cz(X), Z ⊆ B ⊆ U satisfying µ(U\Z) ≤ ε is an algebra. By (7), cz(X) ⊆ B and so Ba(X) ⊆ B. The function F restricted to Ba(X) is a positive Baire measure on X. From (6), it immediately follows that µ is tight. We will now show that T is γ-continuous. [Let {fα }α∈A be a ubounded net in Cb (X, E) which is mt -convergent to 0. There is a δ > 1 such that {fα } ⊆ δN(1, W2 ). Fix ε > 0. Let Zα = {x ∈ X : fα (x) ∈ / εW1 }, Uα = {x ∈ X : fα (x) ∈ / εW2 }. Then Zα ∈ z(X), Uα ∈ cz(X) and Zα ⊆ Uα . We can find functions θα ∈ C(X, I) such that θα ≺ Uα and θα (Zα ) = 1, α ∈ A. Let hα = θα fα and kα = (1 − θα )fα . Then kα ≺ Uα and δ −1 hα ∈ N(1, W2 ), so that |T (hα )| ≤ δµ(Uα ). Moreover, kα (X) ⊆ εW1 , and so kα ∈ sεN(1, W1 ) where s = sup{|ϕ(x)| : x ∈ X}. Thus |T (fα | ≤ |T (hα | + |T (kα | ≤ δµ(Uα ) + sε. This implies that limα T (fα = 0 and so, by Lemma 7.1.2, T is γcontinuous.] ∗

Theorem 7.1.4. The space Cb (X)⊗E is σ(Y, Y )-dense in Cb (X, E), where Y = (Cb (X, E), β). ∗ Proof. By Theorem 7.1.3, σ(Y, Y ) 6 γ. Now, the statement immediately follows from Lemma 7.1.2.

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2. Weak Approximation in CVo (X, E) We now present a “convexified” version of weighted approximation, as given in [Naw89]. Notation. Let SM(X) be the family of all set functions µ : cz(X) −→ [0, ∞], usually called the submeasures, satisfying: (S1) µ(∅) = 0, (S2) µ(U1 ) ≤ µ(U2 ) if U1 ⊆ U2 , (S3) µ(U1 ∪ U2 ) ≤ µ(U1 ) + (U2 ), (S4) for every U ∈ cz(X) and an ε > 0 there is a Z ∈ z(X) such that Z ⊆ U and µ(U\Z) ≤ ε. It easily follows from (S1) - (S4) that the family of all subsets B of X satisfying (∗∗) “for every ε > 0 there are U ∈ cz(X) and Z ∈ z(X) such that Z ⊆ B ⊆ U and µ(U\Z) ≤ ε ” is an algebra containing z(X). Therefore, (∗∗) holds for any set B belonging to Ba(X). The family SM(X) appears naturally while studying continuous linear functionals on CVo (X, E). Indeed, the following result holds. Lemma 7.2.1. [Naw89] Let N(v, W ) be an ωV -neighborhood of 0 in CVo (X, E), where W is balanced. If T is a linear functional on CVo (X, E) which is bounded on N(v, W ), then the function µ defined on cz(X) by µ(U) = sup{|T (f )| : f ∈ N(v, W ), f ≺ U} belongs to SM(X). Proof. Properties (S1) and (S2) are obvious. We shall show that µ satisfies (S3). Fix U1 , U2 ∈ cz(X), ε > 0 and f ∈ N(v, W ) such that supp(f ) ⊆ Z ⊆ U1 ∪ U2 for some Z ∈ z(X). vf vanishes at infinity so there is a compact subset Kε of X such that vf (X\Kε ) ⊆ εW. Let K = Z ∩ Kε . Ui is a cozero set, so we can find a function θi ∈ C(X, I) such that supp(θi ) = Ui , i = 1, 2. K is compact, so inf {(θ1 ∨ θ2 )(x) : x ∈ K} = a > 0. Let 1 1 2 2 θi′ := (θi ∨ aχX ) − aχX , i = 1, 2, and θo := aχX − ( aχX ∧ θ1 ∧ θ2 ). 3 3 3 3 ′ ′ ′ −1 Moreover, let ϕi := θi (θo + θ1 + θ2 ) for i = 0, 1, 2. We define fi := ϕi f for i = 0, 1, 2. Then f = fo + f1 + f2 , fi ∈ N(v, W ), fi ≺ Ui for i = 1, 2 and fo ∈ εN(v, W ). Therefore

2. WEAK APPROXIMATION IN CVo (X, E)

121

|T (f )| ≤ |T (fo )| + |T (f1 )| + |T (f2 )| ≤ εδ + µ(U1 ) + µ(U2 ), where δ = sup{|T (f )| : f ∈ N(v, W )}. This implies (S3). If (S4) fails to be true, then for some ε > 0 we can find, by induction, a sequence {fn } ⊆ N(v, W ) such that supp(fn ) ∩ supp(fm ) = ∅ for n 6= m and T (fn ) > ε, n, m = 1, 2, .... However the functions gn := f1 + ..+ fn belong to N(v, W ) and T (gn > εn for every n ∈ N. This means that T is unbounded on N(v, W ), a contradiction. Notation. If µ ∈ SM(X), we denote by µ∗ the outer measure defined on X by µ; i.e. µ∗ (A) = inf {µ(U) : U ∈ cz(X), χA ≺ U} for A ⊆ X. Definition. Let V be a fixed Nachbin family of weights on X. We define mV -topology as the vector topology on CVo (X, E) which has as a base of neighborhoods of 0 the family of all sets of the form M(µ, v, W, ε) = {f ∈ CVo (X, E) : µ∗ ({x ∈ X : vf (x) ∈ / W }) ≤ ε}, where µ ∈ SM(X), v ∈ V, W ∈ W and ε > 0. Let γV be the strongest vector topology on CVo (X, E) which is weaker than mV on all ωV -bounded subsets of CVo (X, E). Lemma 7.2.2. [Naw89] Let T be such a linear functional on CVo (X, E) that limα T (fα = 0 for every net {fα } ⊆ CVo (X, E) which is ωV -bounded and σV -convergent to 0. Then T is γV -continuous. Proof. It is easily seen that the weakest vector topology on CVo (X, E) for which T is continuous is weaker than mV on every ωV -bounded set. Theorem 7.2.3. [Naw89] For every Nachbin family V , the topology ∗ γV is stronger than the weak topology w = w(Y, Y ) of the space Y = (CVo (X, E), ωV ). Proof. Let us fix an ωV -continuous linear functional T on CVo (X, E). We can find an ωV -neighborhood N(v, W ) of 0 such that W is balanced and |T (f )| ≤ 1 for all f ∈ N(v, W ) Using T and N(v, W ), we define µ ∈ SM(X) as in Lemma 7.2.1; that is µ(U) = sup{|T (f )| : f ∈ N(v, W ), f ≺ U}, U ∈ cz(X).

122


Theorem 7.2.3 will be proved if we show that T is γV -continuous. Let {fα } be a ωV -bounded net and mV -convergent to 0 in CVo (X, E) (cf. Lemma 7.2.3). Fix ε > 0. For every α, there are Uα ∈ cz(X) and Zα ∈ z(X) such that Aα := {x : vf (x) ∈ / εW } ⊆ Uα and µ(Uα ) ≤ µ∗ (Aα ) + ε. There is θα ∈ C(X, I) such that θα ≺ Uα and θα (Zα ) = 1. Let hα := θα fα and kα := (1 − θα )fα . The net {fα } is bounded, so there is a δ > 1 such that {fα } ⊆ δN(v, W ). Thus, δ −1 hα ∈ N(v, W ) and hα ≺ Uα , so that |T (hα )| ≤ δµ(Uα ) ≤ δ(µ∗ (Aα ) + ε). Moreover, kα ∈ εN(v, W ), and so |T (hα )| ≤ δ(µ∗ (Aα ) + ε) + ε. This implies that limα T (fα ) = 0.

Notation. In the sequel we will denote by CVo (X) Pn⊗d E the subset of CVo (X, E) consisting of all functions of the form i=1 θi ⊗ ai , where θi ∈ CVo (X), supp(θi ) ∩ supp(θj ) = ∅ if i 6= j, ai ∈ E, i, j = 1, ..., n, n ∈ N. Theorem 7.2.4. [Naw89] Let V be a Nachbin family of weights on a locally compact Hausdorff space X. For every balanced neighborhood W of 0 in E and v ∈ V , the γV -closure of N(v, 2W ) ∩ [CVo (X) ⊗d E] contains N(v, W ). In particular, CVo (X) ⊗ E is γV and weakly dense in CVo (X, E). Proof. Let us fix v, W and f ∈ N(v, W ). The set Af := {θf ∈ C(X, I), supp(θ) is compact} is ωV -bounded and contained in N(v, W ). Moreover, f can be ωV approximated by elements of Af (see Section 5.2). The topology σV is weaker than ωV , and so γV is weaker than ωV on Af . This implies that f belongs to the γV -closure of Af . Therefore, for the proof of the theorem, we may assume that K = supp(f ) is compact. Let us fix some relatively compact neighborhood UPof K, and let D ⊆ C(X, I) ⊗ E be the set of all functions of the form ni=1 θi ⊗ f (xi ), where θi ∈ C(X, I), supp(θi ) ⊆ U, supp(θi ) ∩ supp(θj ) = ∅, if i 6= j, xi ∈ U, i, j = 1, ..., n, n ∈ N. The set f (U) is bounded in E and each function v ∈ V, being upper semicontinuous, is bounded on U, so D is ωV -bounded. We shall show that f can be mV -approximated by elements of D ∩ N(v, 2W ).


123

[Fix any γV -neighborhood M(µ, v ′ , W ′, ε) of 0, where W ′ is balanced. We may assume that sup{v ′ (x) : x ∈ U} =: a > 0. The set f (K) is compact, so there exists a finite family {B1 , ..., Bn } of pairwise disjoint subsets of X such that K ⊆ ∪n Bi ⊆ U, Bi ∈ B(X), and i=1

1 f (x) − f (y) ∈ W ′ for every x, y ∈ Bi , i = 1, ..., n. a The submeasure µ satisfies (∗∗), so we can choose cozero sets U1 , ..., Un and zero sets Z1 , ..., Zn such that Zi ⊆ Bi ⊆ Ui ⊆ U and µ(Ui \Zi ) ≤ ε/n, i = 1, ..., n. Clearly each Zi is compact. Using the upper semicontinuity of v, we can find pairwise disjoint cozero sets Oi , ..., On such that Zi ⊆ Oi ⊆ Ui and sup{v(x) : x ∈ Oi } < 2 sup{v(x) : x ∈ Zi } for i = 1, ..., n. Choose xi ∈ Zi satisfying sup{v(x) : x ∈ Oi } < 2v(xi ) and θi ∈ C(X, I) such that supp(θi ) = Oi and θi (Zi ) = 1, i = 1, ..., n. Pn We define g = i=1 θi ⊗ f (xi ). Then g ∈ D and v(x)θi (x) < 2v(xi ) for every x ∈ Oi , i = 1, ..., n. Thus g ∈ D ∩ N(v, 2W ). Moreover, n [ ′ ′ / W ′} {x ∈ X : v (f − g)(x) ∈ / W } ⊆ {x ∈ Ui : v ′ (f − g)(x) ∈ i=1

⊆

n [

−1

′

{x ∈ Ui : (f − g)(x) ∈ /a W }⊆

i=1

Therefore,

∗

n [

(Ui \Zi ).

i=1

′

′

µ ({x ∈ X : v (f − g) ∈ /W }≤

n X

µ(Ui \Zi ) ≤ ε,

i=1

and so f − g ∈ M(µ, v ′ , W ′ , ε), as required]. The second assertion immediately follows from the first one and Theorem 7.2.3. Open Problem [Naw89]. It is not known whether Theorem 7.2.4 holds if X is non-locally compact. Theorem 7.2.5. [Naw89] Let V be Nachbin family on a locally compact Hausdorff space X. For every Hausdorff TVS E, the locally convex topology associated with the weighted topology ωV = ωV (X, E) coincides with the topology induced on CVo (X, E) by the weighted topology

124


ωV (X, E c ) of the space CVo (X, E c ), where E c = (E, τ c ) is the locally convex space associated with E = (E, τ ). Proof. The topology τ c is weaker that τ , so that CVo (X, E) ⊆ CVo (X, E c ) and the inclusion mapping is continuous. Therefore, ωVc (X, E) ≥ ωV (X, E c )|CVo (X,E) . For the proof of the converse inequality let us fix a convex weakly closed ωVc (X, E)-neighborhood G of 0 in CVo (X, E). We can find v ∈ V and a balanced neighborhood W of 0 in E such that conv N(v, W ) ⊆ G. The set N := N(v, (1/2) convW ) = {f ∈ CVo (X, E) : vf (X) ⊆ (1/2) convW } is an ωV (X, E c )|CVo (X,E) -neighborhood of 0, which is ωV (X, E)-closed. Therefore, N is closed in the weak topology w of (CVo (X, E), ωV ). Let Nd := N(v, convW ) ∩ [CVo (X) ⊗d E] and where W∞

Nd,∞ := {f ∈ CVo (X) ⊗d E : vf (X) ⊆ W∞ } ∞ S Pn := Wn and Wn := 2−n 2i=1 W . Since w 6 γV , Theorem n=1

7.2.4 shows that W∞

w

w

(1) N = N ⊆ Nd . is τ -dense in conv W , and so Nd,∞ is ωV -dense in Nd . Consequently, Nd,∞ is weakly dense in Nd .

(2)

Let us observe now that Nd,∞ ⊆ conv N(v, W ).

(3)

[Indeed, if f ∈ Nd,∞ then we can find θ1 , ..., θk ∈ CVo (X) and a1 , ..., ak ∈ W∞ such that f =

k X

θi ⊗ ai , supp(θi ) ∩ supp(θj ) = ∅ if i 6= j,

i=1

sup{vθj (x)

:

x ∈ X} ≤ 1, i, j = 1, ..., k.

There are ai,j ∈ W , i = 1, ..., k, j = 1, ..., m, m ∈ N, such that m

1 X ai,,j , i = 1, ..., k. ai = m j=1

P θi ⊗ai,,j , j = 1, ..., m. Then it is easily seen that vfj (X) ⊆ Put fj := ki=1P W and f = m1 m j=1 fj . Therefore, f ∈ conv N(v, W ).]


125

Combining (1), (2) and (3), we have w

w

N ⊆ N d ⊆ N d,∞ ⊆ conv w N(v, W ) ⊆ G. Finally, G is an ωV (X, E c )|CVo (X,E) neighborhood of 0.

Remark 7.2.6. [Naw89] It follows from the above results that ωV (X, E) and ωV (X, E c )|CVo (X,E) produce the same spaces of continuous linear functionals on CVo (X, E). Moreover, CVo (X, E) is dense in CVo (X, E c ). Therefore, we can identify the dual spaces of CVo (X, E) and CVo (X, E c ). Indeed, the restriction mapping CVo (X, E c )∗ ∋ T −→ T |CVo (X,E) ∈ CVo (X, E)∗ is an algebraic isomorphism.

126


3. Notes and Comments Section 7.1. This section contains some results regarding weak approximation in the space (Cb (X, E), β), as given in [Naw85]. This requires the measure theoretic terminology of Section A.7. As seen in Section 4.1, Cb (X) ⊗ E is β-dense in Cb (X, E) for a locally convex E, and also for some concrete classes of not necessarily locally convex spaces E (such as admissible spaces and spaces having the approximation property); the general case of density problem remains open. In connection with the Riesz-type representations of functionals on Cb (X, E), this had been obstructing research in the sense that the representation of functionals was limited to the subspace Cb (X) ⊗ E (see, e.g., [Shu72, KR81, Kat83]). However, using some idea of “submeasure convergence”, Kalton [Kal83] realized that at least if X is compact (and then β is the uniform topology), the “representation theory” can avoid the “density problem”. Later Nawrocki [Naw85] proved that a version of this remains valid also if X is an arbitrary completely regular space. This is used to obtain density results in the weak topology, (i.e. the weakest one defined by the duals of (Cb (X, E), β)). This result is used later in Chapter 8 to obtain a representation of β-continuous linear functionals on Cb (X, E). Section 7.2. This section contains some results regarding weak approximation in the space (CVo (X, E), ωV ), as given in [Naw89]. The main result in [Naw89] states that, if X is locally compact, then CVo (X)⊗ E is always weakly dense in CVo (X, E). This implies that CVo (X) ⊗ E is dense in CVo (X, E) equipped with the locally convex topology ωVc associated to ωV (i.e., the strongest locally convex topology on CVo (X, E) which is weaker than ωV ), where E c is the locally convex space associated to E. Again, this result is required later in Chapter 8 to obtain a representation of ωV -continuous linear functionals on CVo (X, E).

CHAPTER 8

The Riesz Representation type Theorems In this chapter, we first define and establish existence of the integral of functions in Cb (X, E) with respect to certain E ∗ -valued measures, and then obtain Riesz representation type theorems for the characterization of dual spaces (Cb (X, E), u)∗ , (Cpc (X, E), β)∗ , (Crc (X, E), σ)∗ and (CVo (X, E), ωV )∗ .

127

128

8. THE RIESZ REPRESENTATION TYPE THEOREMS

1. Vector-valued Measures and Integration We shall use here the measure theoretic terminology of Section A.7. Let X be a completely regular Hausdorff space and E a real Hausdorff TVS with non-trivial dual E ∗ and having a base W of closed balanced shrinkable neighborhoods of 0 in E. Let z(X) (resp. c(X), k(X)) be the collection of all zero (resp. closed, compact) subsets of X. Let B(X) denote the algebra generated by z(X) and let Ba(X) (resp. Bo(X)) denote the σ-algebra generated by z(X)(resp. c(X)). The space of all Baire measures is denoted by M(Ba(X)), respectively. Let Mσ (Ba(X)) (resp. Mτ (Ba(X)), Mt (Bo(X))) denote the space of all bounded real-valued σadditive (resp. τ -additive, tight) measures on Ba(X) (resp. Bo(X), Bo(X)). Restricting the above measures to Ba(X), we clearly have Mt (Ba(X)) ⊆ Mτ (Ba(X)) ⊆ Mσ (Ba(X)). Note that Mt (Bo(X)) ⊆ Mσ (Ba(X)) need not hold in general as an m ∈ Mτ (Bo(X)) is not necessarily z(X) regular. Every tight m ∈ M(B(X)) is τ -additive and hence it has a unique extension m ˆ ∈ Mτ (Bo(X)). We shall require the following classical versions of the Riesz representation theorem. ∗ Theorem 8.1.1. (a) (Riesz-Alexandroff) (Cb (X), R u) = M(B(X)) via the linear isomorphism L → µ, where L(g) = x gdµ for all g ∈ Cb (X) and ||L|| = |µ|(X). ∗ (b) (Riesz-Buck) (Cb (X), R β) = Mt (Bo(X)) via the linear isomorphism L → µ, where L(g) = x gdµ for all g ∈ Cb (X) and ||L|| = |µ|(X). Proof. (a) See ([Var61] or [Whe83]). (b) See ([Gil71] or [Sen72]).

Definition. For each W ∈ W, let MW (B(X), E ∗ ) denote the set of all finitely-additive E ∗ -valued set functions m on B(X) such that (i) for each a 6= 0 in E, ma (F ) = m(F )(a) (F ∈ B(X)) determines an element ma of M(B(X)); (ii) there exists a constant λ > 0 such that |m|W (X) ≤ λ, where, for each F ∈ B(X), we define |m|W by |m|W (F ) = sup |

X

m(Fi )(ai )|,

i

(it would be the same for a balanced W ) the supremum being taken over all finite partitions {Fi } of F into sets in B(X) (henceforth referred to as a B(X)-partition) and all finite collections {ai } ⊆ W.

1. VECTOR-VALUED MEASURES AND INTEGRATION

Let M(B(X), E ∗ ) =

129

∪ MW (B(X), E ∗ ). Similarly, we may define

W ∈W ∗

the spaces Mσ (Ba(X), E ), Mτ (Bo(X), E ∗ ) and Mt (Bo(X), E ∗ ) by replacing M(B(X)) by Mσ (Ba(X)), Mτ (Bo(X)) and Mt (Bo(X)), respectively, in the above definition. Lemma 8.1.2. [KR81, Kat83] Let W ∈ W. If m ∈ MW (B(X), E ∗ ) (resp. Mσ,W (Ba(X), E ∗ ), Mτ,W (Bo(X), E ∗ ), Mt,W (Bo(X), E ∗ )), then |m|W ∈ M(B(X)) (resp. Mσ (Ba(X)), (Mt (Bo(X)), (Mt (Bo(X))). Proof. First, let m ∈ MW (B(X), E ∗ ). It follows immediately from the definition that |m|W is a bounded non-negative set function on X, and it is straightforward to show that |m|W is finitely additive. To show that |m|W is regular, let F ∈ B(X) and ε > 0. There exist a B(X)-partition {Fi } (1 ≤ i ≤ n) of F and a collection {ai }(1 ≤ i ≤ n) of points in W such that n X |mai | (Fj ) + ε |m|W (F ) ≤ i=1

Since each mai is regular, there exist zero sets Zi (i = 1, ..., n) such that Zi ⊆ Fi and |mai | (Fj ) < |mai | (Zj ) + ε/2i . Let Z = ∪n Zi . Then Z ⊆ F and i=1

|m|W (F ) ≤

n X

|mai | (Zi ) + 2ε ≤ |m|W (F ) + 2ε.

i=1

Thus |m|W ∈ M(X). Now, let m ∈ Mt,W (Bo(X), E ∗ ). Then, by the above argument, it easily follows that |m|W is a bounded, non-negative, finitely-additive set function on Bo(X). We show that |m|W is countably additive, as follows. Let {Ak : k = 1, 2, ....} be a sequence of disjoint sets in Bo(X) and ∞ suppose that ∪ Ak = A. For any n ≥ 1, k=1

n

|m|W (A) ≥ |m|W ( ∪ Ak ) = k=1

n X

|m|W (Ak ),

k=1

and so |m|W (A) ≥

∞ X k=1

|m|W (Ak ).

130


Let ε > 0. Then there exist a Bo(X)-partition {Fj : j = 1, ..., p} of A and a collection of points {aj : j = 1, ..., p} ⊆ W such that |m|W (A) ≤

p X

maj (Fj ) + ε.

j=1

Since each maj is countably additive P and {Fj ∩ Ak : k = 1, 2, ...} is a partition of Fj , we have maj (Fj ) = ∞ k=1 maj (Fj ∩ Ak ). Hence p ∞ ∞ X X X |m|W (A) + ε. maj (Fj ∩ Ak ) + ε ≤ |m|W (A) ≤ j=1 k=1

k=1

Since ε is arbitrary, it follows from the above inequalities that |m|W is countably additive. Next, using the argument of the first part, it easily follows that |m|W is k(W )-regular. Thus |m|W ∈ Mt (Bo(X)). The proof for other parts is similar to the above one. The advantage of taking the base W of ‘shrinkable’ closed balanced neighborhoods is that the Minkowski functional ρW of each W ∈ W is continuous and absolutely homogeneous, and further that W = {x ∈ X : ρW (x) ≤ 1} (see Section A.4). For this reason, the following approach of defining the integral seems to be simpler than those given in [KR81, Kat83] where the notions of F -seminorms and bipolars are used, respectively. Definition. [KR81, Kh95] If f ∈ Cb (X, E) and W ∈ W, we write kf kW = kρW ◦ f k = sup |ρW (f (x))| . x∈X

∗

Let m ∈ MW (B(X), E ), (W ∈ W), and f ∈ Cb (X, E). Let D be the collection of all α = {F1 , ..., Fn ; x1 , ..., xn }, where {Fi } (1 ≤ i ≤ n) is a B(X)-partition of X and xi ∈ Fi . If α1 , α2 ∈ D, define α1 ≥ α2 iff each set which appears in α1 is containedP in some set in α2 . In this way, D becomes as indexing set. Let Sα = ni=1 m(Fi )(f (xi )). We now define the integral of f with respect to m over X by Z dmf = lim Sα . X

α∈D

Regarding the conditions under which this integral exists, we obtain the following lemma. R Lemma 8.1.3. [KR81, Kh95] The integral X dmf , defined above, exists in each of the following cases: (a) f ∈ Cpc (X, E);


131

(b) f ∈ Cb (X, E), and |m|W is τ -additive (in particular, |m|W is tight); (c) f ∈ Cb (X, E), and |m|W is σ-additive, and the range of f is measure compact (i.e. Mσ (f (X)) = Mτ (f (X))). Proof. We need to show that, in each case, {Sα : α ∈ D} is a Cauchy net in R. (a) Let ε > 0. Choose a balanced shrinkable V ∈ W such that V + V ⊆ εW . Since f (X) is precompact, there exist Y1 , ..., Yn ∈ X such n S that f (X) ⊆ (f (yi ) + V ). Let Vi = {x ∈ X : ρV (f (x) − f (yi )) ≤ 1}. i=1

i−1

Then each Vi ∈ B(X). Let G1 = V1 and Gi = Vi \ ∪ Vj (2 ≤ i ≤ n). By j=1

keeping those Gi ’s which are non-empty, we get, {G1 , ..., Gk } say, a B(X)partition of X. Choose xi ∈ Gi and let αo = {G1 , ..., Gk ; x1 , ..., xk }. Note that if x, y are in the same Gi , f (x) − f (y) = [f (x) − f (yi )] + [f (yi ) − f (x)] ∈ V + V ⊆ εW. Then, for α1 , α2 ≥ αo , we have |Sα1 − Sα2 | ≤ |Sα1 − Sαo | + |Sαo − Sα2 | . Suppose α1 = {F1 , ..., Fq ; z1 , ..., zq }, where each Fj is contained in some Gi and zj ∈ Fj . Now q k X X m(Fj )(f (zj )) − m(Gi )(f (xi )) |Sα1 − Sαo | = j=1 i=1 X k X X q m(Fj )(f (zj )) − m(Fj )(f (xi )) = j=1 i=1 j,Fj ⊆Gi X k X −1 m(Fj )[ε (f (zj ) − f (xi ))] ≤ ε |m|W (X). = ε i=1 j,Fj ⊆Gi Similarly, we can prove that |Sα2 − Sαo | ≤ ε |m|W (X). Thus |Sα2 − Sα1 | ≤ 2ε |m|W (X), and {Sα : α ∈ D} is a Cauchy net in R. (b) Let ε > 0, and choose an open balanced shrinkable set V ∈ W with V +V ⊆ εW . The collection V = {f −1(f (y)+V ) : y ∈ X} is a cover of X consisting of cozero sets. Labelling V as {Vλ : λ ∈ I}, we make I into a directed set by saying that λ ≥ γ ⇔ Vλ ⊆ Vγ . By the τ -additivity

132

8. THE RIESZ REPRESENTATION TYPE THEOREMS k

of |m|W , there exist y1 , ..., yk ∈ X such that |m|W (X\ ∪ Vλ1 ) < ε, where i=1

Vλ1 = f

−1

(f (yi ) + V ) = {x ∈ X : ρV (f (x) − f (yi )) < 1}. i−1

k

j=1

i=1

Define G1 = Vλ1 , Gi = (Vλi \ ∪ Vλj (2 ≤ i ≤ k), and Gk+1 = X\ ∪

Vλi . Assuming that Gi ’s are non-empty, choose xi ∈ Gi and let αo = {G1 , ..., Gk+1; x1 , ..., xk+1 }. Let α1 , α2 ≥ αo . Suppose α1 = {F1 , ..., Fq ; z1 , ..., zq }, where each Fj is contained in some Gi and zj ∈ Fj . Then k X X m(Fj )(f (zj ) − f (xi )) + |Sα1 − Sαo | ≤ i=1 j,Fj ⊆Gi X X m(Fj )(f (xk+1 )) m(Fj )(f (zj )) + + j,Fj ⊆Gk+1 j,Fj ⊆Gk+1 ≤ ε(|m|W (X) + 2 kf kW ).

Similarly, we obtain |Sα2 − Sαo | ≤ ε(|m|W (X) + 2 kf kW ). Consequently, {Sα : α ∈ D} is a Cauchy net in R. (c) Suppose |m|W is σ-additive, and let µ(A) = |m|W (f −1 (A)) for every Baire set A of f (X). Then µ is also σ-additive. Now taking V as in part (b) and using the fact that µ is also τ -additive (since f (X) is measure compact), we can complete the proof by the argument of part (b). Remark. Part (b) and (c) were proved in ([Fon74], Lemma 3.11) assuming E a normed space. ∗ Lemma 8.1.4. [KR81] R Let m ∈ MW (X, E ) (W ∈ W) and f ∈ Cb (X, E). If the integral X dmf exists, then Z Z dmf ≤ (ρW ◦ f )d |m|W ≤ kf kW |m|W (X). X

X

Proof. For any ε > 0, There exist a B(X)-partition, {Fi : 1 ≤ i ≤ n} say, of X, and points xi ∈ Fi such that Z n X dmf ≤ m(Fi )(f (xi )) + ε X i=1

and

Z n X (ρW ◦ f )d |m|W + ε. (ρW ◦ f )(xi ) |m|W (Fi ) ≤ X i=1


133

Let H1 (resp. H2 ) be the set of i ∈ {1, ..., n} such that ρW (f (xi )) 6= 0 (resp. ρW (f (xi )) = 0). We note that, if j ∈ H2 , then ρW (tf (xi )) = 0 for all t > 0. Then Z X f (xi ) dmf ≤ (ρW ◦ f )(xi ) m(Fi ) (ρ ◦ f )(x ) W i X i∈H1 X ε |m| (X)f (x ) i W m(Fi ) +ε + |m| (X) ε W i∈H2 X f (xi ) + 2ε. (ρW ◦ f )(xi ) m(Fi ) ≤ (ρ ◦ f )(x ) W i i∈H 1

We note that, if |m|W (X) = 0, then the inequality we are seeking to establish holds trivially. It follows that Z X dmf ≤ (ρW ◦ f )(xi ) |m|W (Fi ) + 2ε X i∈H1 Z ≤ (ρW ◦ f )d |m|W + 2ε, X

and so, since ε is arbitrary, Z Z dmf ≤ (ρW ◦ f )d |m|W . X

X

The other inequality is straightforward to prove.

Note. It is easy to verify that, if m ∈ M(B(X), E ∗ ), g ∈ Cb (X), and a ∈ E, then Z Z dm(g ⊗ a) = gdma . X

X

Remark. An other equivalent approach to define |m|W , due to Katsaras [Kat83], is as follows: For p a continuous seminorm on E, we define Mp (B(X), E ∗ ) to be the set MW (B(X), E ∗ ) where W = Wp = {a ∈ E : p(a) ≦ 1}. For an m ∈ Mp (B(X), E ∗ ), we define mp to be the measure |m|W . Let now W ∈ W be balanced, and let W 00 be the bipolar of W with respect to the pair hE, E ∗ i. Let q = qW be the Minkowski functional of the absolutely convex set W 00 . We have the following: Lemma 8.1.5. [Kat83] A set function m : B(X) → E ∗ belongs to MW (B(X), E ∗ ) iff m ∈ Mq (B(X), E ∗ ). Moreover, we have mq = |m|W . Proof. Since W ⊆ {a ∈ E : q(a) ≦ 1}, it follows that m ∈ MW (B(X), E ∗ ), whenever m ∈ Mq (B(X), E ∗ ), and that |m|W ≦ mq .

134


On the other hand, let m ∈ MW (B(X), E ∗ ) and let A ∈ B(X). For each ϕ ∈ E ∗ , we define |ϕ|W = sup{|ϕ(a)| : a ∈ W }. Then W 0 = {ϕ ∈ E ∗ : |ϕ|W ≦ 1}. It is easy to see that |ϕ|W = sup{|ϕ(a)| : q(a) ≦ 1}. From the definition n X |m|W (A) = sup{ |m(Ai )|W : {Ai }ni=1 a B(X)-partition of A}, i=1

it follows now easily that mq (A) ≦ |m|W (A).

2. INTEGRAL REPRESENTATION THEOREMS

135

2. Integral Representation Theorems We first characterize the u-dual of Cpc (X, E) via the integral representation. Recall that Cb (X) ⊗ E is weakly dense in (Cpc(X, E), u) (see Chapter 7). Theorem 8.2.1. [Kat83, Kh95] (Cpc (X, E), u)∗ = M(B(X), E ∗ ) via the linear isomorphism L → m given by Z L(f ) = dmf (f ∈ Cpc (X, E)). (1) X

Furthermore, if L is represented as in (1) with m ∈ MW (B(X), E ∗ )(W ∈ W), then kLkW = |m|W (X), where kLkW = sup{|L(f )| : f ∈ Cpc (X, E), kf kW ≤ 1}. Proof. Let m ∈ MW (B(X), E ∗ ) (W ∈ W), and suppose that L is the linear functional on Cpc (X, E) defined by (1). Then, by Lemma 8.1.4, |L(f )| ≤ kf kW |m|W (X) ≤ |m|W (X), whenever f ∈ Cpc (X, E) with f (X) ⊆ W . Hence L ∈ (Cpc (X, E), u)∗. We now show that kLkW = |m|W (X). Clearly, kLkW ≤ |m|W (X). For any ε > 0, there exist a B(X)-partition {F1 , ..., Fn } of X and a collection {a1 , ..., an } ⊆ W such that n X m(Fi )(ai ) + ε. |m|W (X) ≤ i=1

There exist zero sets Zi ⊆ Fi such that |mai | (Zi \Fi ) < nε . Since Cb (X) separates the zero sets in X, the closures Z 1 , ..., Z n in βX are pairwise disjoint. Since the cozero sets in βX form a base for open sets, there ˆ2 in βX with Z i ⊆ Uî . Each exist pairwise disjoint cozero sets Uˆ1 , ..., U ˆ Ui ∩ X is a cozero set in X containing Zi ; there is a cozero set Vi in X with î ∩ X and |ma | (Vi \Zi ) < ε . Zi ⊆ Vi ⊆ U i n Choose hP ∈ C (X) with 0 ≤ h ≤ 1, h (Z ) = 1, and hi = 0 on X\Vi . i b i i i n Let h = i=1 hi ⊗ ai . Then h ∈ Cpc (X, E) and khkW ≤ 1. Now n n Z n Z X X X L(h) = L(hi ⊗ ai ) = dmai + hi dmai ; i=1

i=1

Zi

i=1

Vi \Zi

136


hence

n n n X X X m (Z ) + m (F \Z ) ≤ m (F ) ai i ai i i ai i i=1

i=1

i=1

≤ ε + |L(h)| +

n X

|mai | (Vi \Zi )

i=1

≤ kLkW + 2ε.

Thus |m|W (X) ≤ kLkW . Conversely, suppose that L ∈ (Cpc (X, E), u)∗ . Then there exist a W ∈ W and r > 0 such that |L(f )| ≤ r kf kW

(f ∈ Cpc (X, E)).

(2)

For each a 6= 0 in E, let La (g) = L(g ⊗ Cb (X)). By (2), we have |La (g)| ≤ r||g||ρW (a) for all g ∈ Cb (X), and so La ∈ (Cb (X), u)∗. Hence, by R Theorem 8.1.1(a), there exists an ma ∈ M(B(X)) such that La (g) = gdma (g ∈ Cb (X)) and ||La || = |ma |(X). X For each F ∈ B(X), define m(F )(a) = ma (F ) (a ∈ E). Then |m(F )(a)| ≤ |ma |(X) ≤ rρW (a), and consequently m is a finitely additive E ∗ -valued set function on B(X). Next |m|W (X) ≤ r, as follows. Let {F1 , ..., Fp } be a B(X)-partition of X and {a1 , ..., ap } ⊆ W , and let ε > 0. By using the argument as the one used earlier, there exists an h ∈ Cpc (X, E) with khkW ≤ 1 such that p X m(Fi )(ai ) ≤ |L(h)| + 2ε ≤ r + 2ε. i=1

So |m|W (X) ≤ r and hence m ∈ MW (B(X), E ∗ ). P Now for any f = ki=1 fi ⊗ ai (fi ∈ Cb (X), ai ∈ E) in Cb (X) ⊗ E, Z k k Z k Z X X X L(F ) = L(fi ⊗ ai ) = fi dmai = dm(fi ⊗ ai ) = dmf. R

i=1

i=1

X

i=1

X

X

Since x Cb (X) ⊗ E is weakly dense in (Cpc (X, E), u), the above holds for all f ∈ Cpc (X, E). Finally, m is unique, as follows. Suppose there is a µ ∈ M(B(X), E ∗ ) R such that L(f ) = X dµf R for any R for all f ∈ CRpc (X, E). In particular, Rg ∈ Cb (X) and a ∈ E, X dm(g ⊗ a) = X dµ(g ⊗ a). Hence X gdma = gdµa for all g ∈ Cb (X), and so by Theorem 8.1.1 (a), ma = µa . Thus, X for any F ∈ B(X) and any a ∈ E, m(F )(a) = ma (F ) = µa (F ). It follows that m(F ) = µ(F ), and so m = µ.


137

The following representation theorem gives a characterization for the dual of (Cb (X, E), β). Theorem 8.2.2. [KR81, Kat83] (Cb (X, E), β)∗ = Mt (Bo(X), E ∗ ) via the linear isomorphism L → m given by Z L(f ) = dmf (f ∈ Cb (X, E)). (3) X

Further, if L is represented as in (3) with m ∈ Mt,W (Bo(X), E ∗ ) (W ∈ W), then kLkW = |m|W (X). Proof. Let m ∈ Mt,W (Bo(X), E ∗ ) (W ∈ W), and suppose that L is defined by (3). Then, by Lemma 8.1.2, |m|W ∈ Mt (Bo(X)). It follows from Theorem 8.1.1 that the equation Z LW (g) = gd |m|W (g ∈ Cb (X)). (4) X

defines a β-continuous linear functional LW on Cb (X). Hence there exists a ϕ ∈ Bo (X), 0 ≤ ϕ ≤ 1, such that |LW (g)| ≤ 1 whenever g ∈ Cb (X) with kϕgk ≤ 1. For any f ∈ N(ϕ, W ), ||ϕ(ρW ◦ f )|| = ||ρW (ϕf )|| ≤ 1 and so, by (4) and Lemma 8.1.4, Z |L(f )| ≤ (ρW ◦ f )d |m|W = LW (ρW ◦ f ) ≤ 1. X

Thus L is β-continuous. Conversely, let L ∈ (Cb (X, E), β). Then there exists a θ ∈ Bo (X) and W ∈ W such that |L(f )| ≤ 1 for all f ∈ N(θ, W ).

(5)

For each a 6= 0 in E, let La (g) = L(g ⊗ a) (g ∈ Cb (X)). It easily follows from (5) that La ∈ (Cb (X), β)∗ and so by Theorem 8.1.1, there exists a unique ma ∈ Mt (X) such that Z La (g) = gdma (g ∈ Cb (X)). X

For each F ∈ Bo(X), m(F ) (a) = ma (F ) (a ∈ E). Since β ≤ u, L is u-continuous and so there exists a W ∈ W such that |L(f )| ≤ 1 whenever f ∈ N(1, W ). Consider g ∈ Cb (X) with ||g|| ≤ 1. Then ρW (g(x)a) ≤ ρW (a) for all x ∈ X, and so g ⊗ a ∈ N(1, W ) whenever ρW (g(x)a) ≤ ρW (a) for all a ∈ X. Thus ||La || ≤ ρW (a), and so from the inequalities |m(F )(a)| ≤ |ma (F )| ≤ ||ma || = ||La || ≤ ρW (a),

138


the continuity of m(F ) follows. Consequently m is a finitely additive E ∗ -valued set function on Bo(X) such that, for each a 6= 0 in E, ma ∈ Mt (X). Next |m|W (X) < ∞, as follows. Let {Fj : j = 1, ..., p} be a Bo(X)partition of X and a1 , ..., ap ∈ W , and let ε > 0. Each ma is regular and so there exist compact sets Kj ⊆ Fj with |maj |(Fj \Kj ) < ε/2p, and open sets Vj ⊇ Kj with |maj (Fj \Kj )| < ε/2p for j = 1, ..., p; since Kj ’s are disjoint compact sets and X is completely regular, the Vj ’s may be chosen pairwise disjoint. Choose functions gj (1 ≤ j ≤ p) in P Cb (X), 0 ≤ gj ≤ 1, such that gi = 1 on Kj and supp(gj ) ⊆ Vj . Let h = pi=1 gj ⊗ aj . Then h ∈ Cpc (X, E) and khkW ≤ 1, and so |L(h)| ≤ 1. By using the above inequalities as in the proof of Theorem 8.2.1, we have |m|W (X) ≤ kLkW . Thus m ∈ Mt (Bo(X), E ∗ ). P Consider any f = ki=1 fi ⊗ ai (fi ∈ Cb (X), ai ∈ E) in Cb (X) ⊗ E. Then Z k k k Z X X X L(f ) = L(fi ⊗ ai ) = fi dmai = dm(fi ⊗ ai ) = dmf. i=1

i=1

i=1

X

X

Since Cb (X) ⊗ E is weakly dense in (Cb (X, E), β), the above holds for all f ∈ Cb (X, E). The proof for uniqueness of m and for kLkW = |m|W (X) are the same as in Theorem 8.2.1 with slight modifications. We next characterize the dual of (Crc (X, E), σ), σ being the σ-compactopen topology. Recall that Crc (X, E) denotes the subspace of Cb (X, E) which consists of those functions f such that f (X) is relatively compact. Each f in Cb (X) or Crc (X, E) has a continuous extension fb to βX, the Stone-Cech compactification of X. Let S denote the set of all σ-compact subsets of X and, for any A ∈ S, let A¯ denote the βX-closure of A. Let ¯ E ∗ ) (W ∈ W) denote the set of all m ∈ Mt,W (Bo(βX), E ∗) Mt,W (Bo(A), ¯ let such that the support of |m|W is contained in some A; ¯ E ∗ ) = ∪ Mt,W (Bo(A), ¯ E ∗ ). Mt (Bo(A), W ∈W

¯ E ∗ ) via Theorem 8.2.3. [KR91] (Crc (X, E), σ)∗ = ∪ Mt,W (Bo(A), A∈S

the linear isomorphism L → m given by Z L(f ) = dmfb (f ∈ Crc (X, E)). βX

Moreover, for some W ∈ W, kLkW = |m|W (βX), where kLkW = {|L(f )| : f ∈ Crc (X, E), kρW ◦ f kX ≤ 1}.


139

Proof. Using the Gulick’s version [Gu72] of the Riesz representation ¯ the proof follows as in the theorem that (Cb (X), σ)∗ = ∪ Mt (Bo(A)), A∈S

above theorem with appropriate modifications. The details may be found in ([KR91], Theorem 4.4). Finally, we state a characterize the dual of the weighted space (CVo (X, E), ωV ). Here we follow the argument due to Nawrocki [Naw89]. We recall from Remark 7.2.6 that the topologies ωV (X, E) and ωV (X, E c )|CVo (X,E) produce the same spaces of continuous linear functionals on CVo (X, E). Moreover, CVo (X, E) is dense in CVo (X, E c ). Therefore, we can identify the dual spaces of CVo (X, E) and CVo (X, E c ). Indeed, the restriction mapping CVo (X, E c )∗ T −→ T |CVo (X,E) ∈ CVo (X, E)∗ is an algebraic isomorphism. Therefore, using Theorem 7.2.5 and the known representation of the dual space of CVo (X, E), where E is locally convex (see [Pro71b] or ([Pro77], Theorem 5.42)), we obtain the following result. Proposition 8.2.4. [Naw89] Let X be a locally compact space, V any Nachbin family on X and E a TVS with E ∗ separates the points of E. The dual space of (CVo (X, E), ωV ) is isomorphic to V Mt (Bo(X), E ∗ ) ≡ {vm : v ∈ V, m ∈ Mt (Bo(X), E ∗ ), the space of all (E ∗ , σ(E ∗ , E))-valued bounded tight measures on X.

140


3. Notes and Comments Section 8.1. Various classes of E ∗ -valued measures on a topological space X are considered. The existence of the integral of functions in Cb (X, E) with respect to these measures is also given. These results are taken from the papers [Shu72b, KR81, Kat83, Kh95]. Earlier results in the locally convex setting were obtained in [Kat74, Fon74]. Section 8.2. The Riesz representation type theorems for the characterization of dual spaces (Cb (X, E), u)∗, (Cpc (X, E), β)∗ , (Crc (X, E), σ)∗ and (CVo (X, E), ωV )∗ are obtained via the integral representations. In fact, each of these dual spaces is identified with a suitable class of E ∗ valued measures on X. These results are also taken from the papers [Shu72b, KR81, Kat83, Kh95]. Theorem 8.2.2 has been obtained, independently, in [KR81, Kat83]. For earlier work in the locally convex setting, several useful references can be found in [Shu72b]. Theorem 8.2.4 was obtained in [SumW70] for E a scalar field, in [Pro71b] for E a locally convex space and in [Naw89] for E a general TVS. There is an extensive work (not considered here) on Bochner and Bartle–Dunford–Schwartz integration processes for vector-valued functions and also on the Riesz representation of operators T : Co (X, E) → E for E a Banach or locally convex space (see [Mez02, Mez08, Khu08]). More recent developments in this area have been made in the papers [DL13, Thom12]. These latter papers deal with the theory of integration of scalar functions with respect to a measure with values in a TVS E, not necessarily locally convex. They focus on the extension of such integrals from bounded measurable functions to the class of integrable functions, proving adequate convergence theorems, and establishing usable integrability criteria.

CHAPTER 9

Weighted Composition Operators In this chapter, we give some characterizations of multiplication and composition operators on the weighted function spaces CVo (X, E) and CVb (X, E) induced by scalar-, vector-, and operator-valued maps. We also include characterizations of, compact, equicontinuous, precompact and bounded multiplication/composition operators. As before, the topology of bounded convergence tu (resp. the topology of pointwise convergence tp ) on CL(E) is the linear topology which has a base of neighborhood of 0 consisting of all sets of the form U(D, G) = {T ∈ CL(E) : T (D) ⊆ G} = {T ∈ CL(E) : kT kD,G ≤ 1}, where D is a bounded (respectively, finite) subset of E, and G ∈ WE is shrinkable. (Here kT kD,G = sup{ρG (T (a)) : a ∈ D}.)

141

142

9. WEIGHTED COMPOSITION OPERATORS

1. Multiplication Operators on CVo (X, E) Let X be a completely regular Hausdorff space, E a Hausdorff TVS, and V a Nachbin family of weights on X. It is assumed throughout that the Nachbin family V on X satisfies the following conditions: (∗) V > 0; (∗∗) CVo (X) does not vanish on X, i.e., corresponding to each x ∈ X, there exists an hx ∈ CVo (X) such that hx (x) 6= 0. The above conditions primarily serve to exclude trivial cases along with certain unnecessary pathological situations. If V > 0, then condition (∗∗) holds, in particular, when each v ∈ V vanishes at infinity or X is locally compact. [First suppose that V ⊆ So+ (X), and let x ∈ X. Choose v ∈ V such that v(x) 6= 0. Since X is completely regular, there exists an f ∈ Cb (X) such that f (x) = 1. Since v ∈ So+ (X), it easily follows that vf ∈ CVo (X) and v(x)f (x) 6= 0. Next, suppose that X is locally compact, and let x ∈ X. There exists an f ∈ Coo (X) ⊆ Co (X) such that f (x) = 1. Since V > 0, choose v ∈ V such that v(x) 6= 0. Then clearly vf ∈ CVo (X) and v(x)f (x) 6= 0.] We further mention a useful fact that if CVo (X) does not vanish on X, then, for any x ∈ X and any open neighborhood U of x in X, we can choose a f ∈ CVo (X) with 0 ≤ f ≤ 1, f (X\U) = 0 and f (x) = 1. This follows from Lemma 4.1.2, due to Nachbin [Nac67], by taking M = CVo (X) ⊆ C(X), K = {x}, and U = Ui for all i = 1, ..., n. If CVo (X) does not vanish on X and E is a non-trivial TVS, then CVo (X) ⊗ E and hence CVb (X, E) and CVb (X, E) also do not vanish on X. [In fact, for any x ∈ X, choose ϕ in CVo (X) with ϕ(x) 6= 0. Then, if a (6= 0) in E, the function ϕ ⊗ a belongs to CVo (X) ⊗ E and clearly (f ⊗ a)(x) = ϕ(x)a 6= 0.] Note. Earlier in Section 1.2, we have seen by an example that CVb (X, E) and hence CVo (X, E) may be trivial for relatively nice X (e.g. X = Q). This situation justifies to assume the above conditions (∗) and (∗∗) throughout. Definition. Let F (X, E) be the vector space of all functions from X into E. If θ : X → C and ψ : X → E, E a topological algebra, are given maps, the scalar multiplication on E and the multiplication on E give rise to linear mappings Mθ and Mψ from CVo (X, E) into F (X, E) defined by Mθ (f ) = θf and Mψ (f ) = ψf , both pointwise; that is, (θf )(x) = θ(x)f (x) and (ψf )(x) = ψ(x)f (x), x ∈ X.

1. MULTIPLICATION OPERATORS ON CVo (X, E)

143

Also if π : X → CL(E), E a TVS, we may define Mπ from CVo (X, E) into F (X, E) by Mπ (f ) = πf , where (πf )(x) = πx (f (x)) := π(x)(f (x)), x ∈ X. If Mθ , Mψ and Mπ map CVb (X, E) (resp. CVo (X, E)) into itself and are continuous, they are called multiplication operators on CVb (X, E)(resp. CVo (X, E)) induced by θ, ψ and π, respectively. We first give necessary and sufficient conditions for Mθ , θ : X → C, to be the multiplication operator on the weighted space CVo (X, E). These results hold also for the space CVb (X, E) with slight modification in the proofs and are therefore omitted. For any θ : X → C, we let V |θ| = {v|θ| : v ∈ V }. Theorem 9.1.1. [SM91, KT97] For a mapping θ : X → C, the following are equivalent: (a) Mθ is a multiplication operator on CVo (X, E); (b) (i) θ is continuous and (ii) V |θ| ≤ V. Proof. Let W be a base of closed, balanced and shrinkable neighborhoods of 0 in E. (a) ⇒ (b) To prove (i), let {xα } be a net in X with xα → x ∈ X. By assumption (∗∗), there exists an h ∈ CVo (X) such that h(x) 6= 0. Since Mθ is a self-map on CVo (X, E), it follows that the function θh from X into C is continuous. Hence θ(xα )h(xα ) → θ(x)h(x) and consequently θ(xα ) → θ(x). (ii) To show that V |θ| ≤ V , let v ∈ V . By continuity of Mθ , given G ∈ W, there exist u ∈ V and H ∈ W such that Mθ (N(u, H)) ⊆ N(v, G).

(1)

Without loss of generality, we may assume that G∪H is a proper subset of (a) E. Choose a ∈ E\(G ∪ H), and put t = ρρHG (a) . We claim that v|θ| ≤ 2tu. Fix xo ∈ X. We shall consider two cases: u(xo ) 6= 0 and u(xo ) = 0. Suppose that u(xo ) 6= 0, and let ε = u(xo ). Then D = {x ∈ X : u(x) < 2ε} is an open neighborhood of xo . Taking M = CVo (X) with assumption (∗∗) in Lemma 4.1.2, there is an h ∈ CVo (X) with 0 ≤ h ≤ 1, h(xo ) = 1 and h(X\D) = 0. Define f =

1 (h 2ερH (a)

⊗ a). Since ρH is homogeneous, for any x ∈ X,

ρH [u(x)f (x)] = ρH [u(x)

u(x)h(x) 1 h(x)a)] = . 2ερH (a) 2ε

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If x ∈ D, u(x) < 2ε and if x ∈ X\D, h(x) = 0; hence 2εh(x) ≤ 1 if x ∈ D, 2ε ρH [u(x)f (x)] = 0 < 1 if x ∈ X\D.

ρH [u(x)f (x)]
0. Put ε=

1 v(xo )|θ(xo )|. 2t

Let D = {x ∈ X : u(x) < ε}, and choose an h ∈ CVo (X) with 0 ≤ h ≤ 1, h(xo ) = 1 and h(X\D) = 0. Define g = ερH1(a) (h ⊗ a). Since ρH is homogeneous, for any x ∈ X, ρH [u(x)g(x)] = ρH [u(x)

u(x)h(x) 1 h(x)a)] = . ερH (a) ε

If x ∈ D, u(x) < ε and if x ∈ X\D, h(x) = 0; hence εh(x) ≤ 1 if x ∈ D ε ρH [u(x)f (x)] = 0 < 1 if x ∈ X\D.

ρH [u(x)f (x)]
0. Put ε = v(xo )ρG (ψ(xo )) . Let D = {x ∈ X : u(x) < ε}, and choose an h ∈ CVo (X) 2ρH (e) as above. Define g = ερH1(e) (h ⊗ e). Then g ∈ N(u, H) and so by (2), φg ∈ N(v, G). From this we obtain v(xo )ρG (ψ(xo )) ≤ ρH (e)ε = v(xo )ρG (ψ(xo ))/2, which is impossible unless v(xo )ρG (ψ(xo )) = 0. (b) ⇒ (a) We first show that Mψ maps CVo (X, E) into itself. Let f ∈ CVo (X, E), and let v ∈ V and G ∈ W. Choose u ∈ V such that vρG ◦ ψ ≤ u. There exists a compact set K ⊆ X such that u(x)f (x) ∈ G for all x ∈ X\K. Since ρG is submultiplicative, for any x ∈ X\K, ρG (v(x)ψ(x)f (x)) ≤ v(x)ρG (ψ(x))ρG (f (x)) ≤ u(x)ρG (f (x)) ≤ 1; hence Mψ (f ) ∈ CVo (X, E). Using again the submultiplicity of ρG , the continuity of Mψ follows in the same way as in the proof of Theorem 9.1.1. We mention that Theorem 9.1.2 also remains true if we replace CVo (X, E) by CVb (X, E). + We now apply the above results to the cases: V = Soo (X) and V = + So (X). Theorem 9.1.3. [SM91, KT97] (i) If θ : X → C is a continuous mapping, then Mθ is a multiplication operator on (C(X, E), k). (ii) If E is a locally idempotent topological algebra with identity e and ψ : X → E a continuous mapping, then Mψ is a multiplication operator on (C(X, E), k). Proof. (i) In view of Theorem 9.1.1, we only need to verify that + V |θ| ≤ V , where V = Soo (X). Let v ∈ V . Choose a compact set K ⊆ X with v(x) = 0 for all x ∈ X\K. Let s = sup{|θ(x)| : x ∈ K}, t = sup{v(x) : x ∈ K} and u = stχK . Then u ∈ V and clearly v(x) |θ(x)| ≤ u(x) for all x ∈ X. (ii) Let W be a base of neighborhoods of 0 in E consisting of closed, balanced, shrinkable and idempotent sets. In view of Theorem 9.1.2, we only need to verify that V ρG ◦ ψ ≤ V for every G ∈ W, + where V = Soo (X). Let v ∈ V and G ∈ W. Choose a compact set K ⊆ X with v(x) = 0 for all x ∈ X\K. Let s = sup{ρG (ψ(x)) : x ∈


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K}, t = sup{v(x) : x ∈ K} and u = stχK . Then u ∈ V and clearly v(x)ρG (ψ(x)) ≤ u(x) for all x ∈ X. The following example shows that the above result need not hold when (C(X, E), k) is replaced by (Cb (X, E), β). Example 9.1.4. Let X = R+ , E = C, and V = So+ (X). Let θ = ψ : X → C be given by θ(x) = x2 (x ∈ X), and let v ∈ V be given by v(x) = x1 (x ∈ X). Then v(x) |θ(x)| = x for all x ∈ X. Since each u ∈ V is a bounded function, v |θ| ≤ u is not true for every u ∈ V ; i.e.V |θ| ≤ V does not hold and so, by Theorem 9.1.1, Mθ is not a multiplication operator on (Cb (X), β). The same is also true for the space (Cb (X), u), where u is the uniform topology, since β ≤ u. However, if θ and ψ are “bounded” continuous functions, then it is easily seen that Mθ and Mψ are always multiplication operators on CVo (X, E) for any Nachbin family V . We now consider multiplication operators induced by operator-valued mappings and give necessary and sufficient conditions for Mπ , π : X → CLu (E), to be the multiplication operator on the weighted space CVb (X, E). We then use this characterization to show that, in the case of E a locally bounded space, Mπ is a multiplication operator if π is a bounded + mapping or V = Soo (X). Theorem 9.1.5. [SM91, KT97] Let X be a k-space and E a TVS. Then, for any continuous mapping π : X → CLu (E), the following are equivalent: (a) Mπ is a multiplication operator on CVb (X, E). (b) For every v ∈ V and G ∈ W, there exist u ∈ V and H ∈ W such that v(x)ρG (πx (y)) ≤ u(x)ρH (y) for all x ∈ X and y ∈ E. Proof. We may assume that W consists of closed, balanced and shrinkable sets. (a) ⇒ (b) Let v ∈ V and G ∈ W. By hypothesis, there exist u ∈ V and H ∈ W such that Mπ (N(u, H) ⊆ N(v, G).

(3)

We claim that v(x)ρG (πx (y) ≤ 2u(x)ρH (y) for all x ∈ X and y ∈ E. Let xo ∈ X and yo ∈ E. Then we consider four cases: (I)

u(xo )ρH (yo ) 6= 0.

(II)

u(xo ) = 0, ρH (yo ) 6= 0.

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(III)

u(xo ) 6= 0, ρH (yo ) = 0.

(IV)

u(xo ) = 0, ρH (yo ) = 0.

Case I. Suppose u(xo )ρH (yo ) 6= 0, and let ε = u(xo )ρH (yo ). Then the set D = {x ∈ X : u(x)ρH (yo ) < 2ε} is an open neighborhood of xo , and so there exists an h ∈ CVb (X) such that 0 ≤ h ≤ 1, h(xo ) = 1, and 1 h(X\D) = 0. Define g = 2ε (h ⊗ yo). Then, for any x ∈ X, 1 u(x)h(x)ρH (yo ) ≤ 1; 2ε that is, g ∈ N(u, H). Hence, by (3), Mπ (g) ∈ N(v, G). This implies that, for any x ∈ X, ρH (u(x)g(x)) =

v(x)h(x)ρG (πx (yo)) ≤ 2u(xo )ρH (yo). In particular, by taking x = xo , we have v(x)ρG (πxo (yo )) ≤ 2u(xo )ρH (yo ). Case II. Suppose u(xo ) = 0, ρH (yo ) 6= 0, but v(xo )ρG (πxo (yo)) > 0. Put ε = v(xo )ρG (πxo (yo )/2. Then the set D = {x ∈ X : u(x)ρH (yo ) < ε} is an open neighborhood of xo , and so there exists an h ∈ CVb (X) such that 0 ≤ h ≤ 1, h(xo ) = 1, and h(X\D) = 0. Define g = 1ε (h ⊗ yo). Then, for any x ∈ X, 1 ρH (u(x)g(x)) = u(x)h(x)ρH (yo) ≤ 1; ε that is, g ∈ N(u, H). Hence, by (3), Mπ (g) ∈ N(v, G). This implies that, for any x ∈ X, 1 v(x)h(x)ρG (πx (yo )) ≤ ε = v(x)ρG (πxo (yo )). 2 In particular, by taking x = xo , we have 1 v(x)ρG (πxo (yo )) ≤ v(x)ρG (πxo (yo )), 2 which is impossible unless v(xo )ρG (πxo (yo )) = 0. (III). Suppose u(xo ) 6= 0, ρH (yo ) = 0, but v(xo )ρG (πxo (yo)) > 0. Put ε = 21 v(xo )ρG (πxo (yo )). Then the set D = {x ∈ X : u(x) < ε + u(xo )} is an open neighborhood of xo , and so there exists an h ∈ CVb (X) such


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that 0 ≤ h ≤ 1, h(xo ) = 1, and h(X\D) = 0. Define g = h ⊗ yo Clearly g ∈ CVb (X, E) and 0 ≤ ρG ◦ g ≤ ρG (yo), ρG ◦ g(xo ) = ρG (yo ), and ρG ◦ g(X\D) = 0. Consider g1 = gε . Then, for any x ∈ X, 1 1 ρH (g1 (x)) = ρH ( h(x)yo ) = h(x)ρH (yo ) = 0 ∈ H ε ε Hence g1 ∈ N(u, H) and so, by (3), Mπ (g1 ) ∈ N(v, G). Therefore, for any x ∈ X 1 v(x)ρG (πx (g1 (x))) ≤ 1, or v(x)ρG (πx (h(x)yo )) ≤ ε = v(xo )ρG (πxo (yo )). 2 1 In particular, since h(xo ) = 1, v(xo )ρG (πxo (yo )) ≤ 2 v(xo )ρG (πxo (yo )), a contradiction. (IV) Suppose u(xo ) = 0, ρH (yo ) = 0, but v(xo )ρG (πxo (yo )) > 0. Put ε = 21 v(xo )ρG (πxo (yo )). Then the set D = {x ∈ X : u(x) < ε} is an open neighborhood of xo , and so there exists an h ∈ CVb (X) such that 0 ≤ h ≤ 1, h(xo ) = 1, and h(X\D) = 0. Define g = h ⊗ yo. Clearly g ∈ CVb (X, E) and, for any x ∈ X, 0 ≤ ρG (g(x)) = h(x)ρG (yo ) ≤ ρG (yo ), ρG (g(xo )) = ρG (yo), and ρG (g(x)) = 0 if x ∈ X\D; or 0 ≤ ρG ◦ g ≤ ρG (yo), ρG ◦ g(xo) = ρG (yo ), and ρG ◦ g(X\D) = 0. Consider g1 = gε . Then, for any x ∈ X, 1 1 ρH (g1 (x)) = ρH ( h(x)yo ) = h(x)ρH (yo ) = 0 ∈ H ε ε Hence g1 ∈ N(u, H) and so, by (4), Mπ (g1 ) ∈ N(v, G). Therefore, for any x ∈ X 1 v(x)ρG (πx (g1 (x))) ≤ 1, or v(x)ρG (πx (h(x)yo )) ≤ ε = v(xo )ρG (πxo (yo )). 2 1 In particular, since h(xo ) = 1, v(xo )ρG (πxo (yo )) ≤ 2 v(xo )ρG (πxo (yo )), a contradiction. (b) ⇒ (a) We first show that Mπ maps CVb (X, E) into itself. Let f ∈ CVb (X, E). Since X is a k-space, it is enough to show that Mπ (f ) is continuous on any given compact subset K of X. Let {xα : α ∈ I} be a net in K with xα → x ∈ K. Let G ∈ W, and let G1 ∈ W with G1 + G1 ⊆ G. Now {f (xα )}, being convergent, is bounded and so, by continuity of π, there exists an α1 ∈ I such that πxβ (f (xα )) − πx (f (xα )) ∈ G1 ,

(4)

150


for all α ∈ I and all β ≥ α1 . Since πx : E → E is a continuous linear mapping, there exists an H ∈ W such that πx (y) ∈ G1 for all y ∈ H. Choose an α2 ∈ I such that f (x) ∈ H for all α ≥ α2 ; consequently, πx (f (xα ) − f (x)) ∈ G1

(5)

for α ≥ α2 . Choose αo ∈ I with αo ≥ α1 , α2 . Then, by (4) and (5) πxβ (f (xβ )) − πx (f (x)) = πxβ (f (xβ )) − πx (f (xβ ) + πx (f (xβ )) + πx (f (x) ∈ G1 + G1 ⊆ G for all β ≥ αo . Hence Mπ (f ) ∈ C(X, E). To show that Mπ (f ) ∈ CVb (X, E), let v ∈ V and G ∈ W. By hypothesis, there exist u ∈ V and H ∈ W such that v(x)ρG (πx (y)) ≤ ρH (u(x)(y)

(6)

for all x ∈ X and y ∈ E. Choose λ > 0 such that u(x)f (x) ∈ λH for all x ∈ X. Then, it follows from (6) that v(x)πx (f (x)) ∈ λG for all x ∈ X. This proves that Mπ (f ) ∈ CVb (X, E). To establish the continuity of Mπ , let {fα : α ∈ I} be a net in CVb (X, E) with fα → 0, and let v ∈ V and G ∈ W. Choose u ∈ V and H ∈ W as above and which satisfy (6). There exists an αo ∈ I such that fα ∈ N(u, H) for all α ≥ αo . Then, for any x ∈ X and α ≥ αo v(x)ρG (πx (fα (x))) ≤ ρH (fα (x)) ≤ 1 or equivalent v(x)πx (fα (x)) ∈ G. So Mπ is continuous at 0 and hence, by its linearity, on CVb (X, E). We now deduce that, under same additional conditions on π, Mπ is automatically a multiplication operator. Corollary 9.1.6. [SM91, KT97] Let π : X → CLu (E) be a constant mapping. Then Mπ is a multiplication operator on CVb (X, E). Proof. We need to verify that condition (b) of Theorem 9.1.5 holds in this case. Let v ∈ V and G ∈ W. Choose T ∈ CL(E) such that π(x) = T for all x ∈ X. Since T is continuous and linear, there exist an m > 0 and a closed shrinkable H ∈ W such that v(x)ρG (T (y)) ≤ mρH (y) for all y ∈ E. Let u = mv. Then, for any x ∈ X and y ∈ E, v(x)ρG (π(x)(y)) = v(x)ρG (T (y)) ≤ u(x)ρH (y).


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Corollary 9.1.7. Suppose E is a locally bounded TVS and π : X → CLu (E) is a continuous and bounded mapping. Then Mπ is a multiplication operator on CVb (X, E). Proof. Let v ∈ V and G ∈ W. We may assume that G is a bounded neighborhood of 0. Choose a closed and shrinkable H ∈ W with H ⊆ G. Since π(X) is bounded in CLu (E), there exists an m > 0 such that π(X) ⊆ mU(H, H); i.e., ρH (π(x)(y)) ≤ mρH (y) for all x ∈ X, y ∈ H. Let u = mv. Then, for any x ∈ X and y ∈ E, v(x)ρG (π(x)(y)) ≤ v(x)ρH (π(x)(y)) ≤ u(x)ρH (y). + We now apply Theorem 9.1.5 to the cases V = Soo (X) and V = + So (X). Theorem 9.1.8. [SM91, KT97] Suppose E is a locally bounded TVS and π : X → CLu (E) is a continuous mapping. Then Mπ is a multiplication operator on (C(X, E), k). Proof. We need to verify that condition (a) of Theorem 9.1.5 holds + for V = Soo (X). Let v ∈ V and G ∈ W. Choose a compact K ⊆ X such that v(x) = 0 for all x ∈ X\K. We may assume that G is a bounded neighborhood of 0. Choose a closed and shrinkable H ∈ W with H ⊆ G. Since π : X → CLu (E) is continuous, π(K) is compact in CLu (E) and so there exists an m > 0 such that π(K) ⊆ mU(H, H); i.e., ρH (π(x)(y)) ≤ mρH (y) for x ∈ K and y ∈ H. Let t = sup{v(x) : x ∈ K}, and put u = mtχK . Then u ∈ V . Let x ∈ X and y ∈ E. If x ∈ K, then v(x)ρG (π(x)(y)) ≤ tρH (π(x)(y)) ≤ u(x)ρH (y). If x ∈ X\K, the above holds trivially (since v(x) = 0). Remark. If E = C, then CLu (E) = C and so Example 9.1.4 also shows that the above result need not hold when (C(X, E), k) is replaced by (Cb (X, E), β).

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2. Weighted Composition Operators on CVo (X, E) In this section, we consider the more general notion of weighted composition operators which include the multiplication and composition operators. Definition. Let F (X, E) be the vector space of all functions from a topological space X to a TVS E, and let S(X, E) be a vector subspace of F (X, E). Let L(E) be the vector space of all linear maps from E into itself. Then the operator-valued mapping π : X → L(E) and the selfmap ϕ : X → X give rise to a linear mapping Wπ,ϕ : S(X, E) → F (X, E) defined as Wπ,ϕ (f ) = π · f ◦ ϕ, f ∈ S(X, E); that is, Wπ,ϕ (f )(x) = π(x)(f (ϕ(x))) = πx (f (ϕ(x))), f ∈ S(X, E), x ∈ X. (i) In case S(X, E) is equipped with a linear topology, Wπ,ϕ takes S(X, E) into itself and Wπ,ϕ is continuous, we call Wπ,ϕ the weighted composition operator on S(X, E) induced by the mappings π and ϕ. (ii) If ϕ : X → X is the identity map, the corresponding operator Wπ,ϕ is the multiplication operator Mπ on S(X, E) induced by the operator-valued mapping π. (iii) Also, if π(x) = I, the identity operator on E, for every x ∈ X, then Wπ,ϕ is the composition operator Cϕ on S(X, E) induced by the self-map ϕ of X, where Cϕ (f ) = f ◦ ϕ, f ∈ S(X, E). These operators have been the subject matter of extensive study for the last several decades on different functions spaces; especially on Lp spaces, Bergman spaces, and spaces of continuous functions. As before, it is assumed that the Nachbin family V satisfies (∗) V > 0 and (∗∗) CVo (X) does not vanish on X. Theorem 9.2.1. [MS98] Let X be a completely regular kR -space and E a TVS. Let π : X → CLu (E) and ϕ : X → X be continuous mappings. Then the following conditions are equivalent: (a) Wπ,ϕ is a weighted composition operator on CVo (X, E). (b) (i) for any v ∈ V and G ∈ W, there exist u ∈ V and H ∈ N such that u(ϕ(x))y ∈ H implies that v(x)πx (y) ∈ G for all x ∈ X, y ∈ E; (ii) for each v ∈ V and G ∈ W and compact set K ⊆ X, the set ϕ−1 (K) ∩ {x ∈ X : v(x)πx (y) ∈ / G} is compact for each 0 6= y ∈ E.

2. WEIGHTED COMPOSITION OPERATORS ON CVo (X, E)

153

Proof. (a) ⇒ (b) Suppose that Wπ,ϕ is a weighted composition operator on CVo (X, E). To establish condition (b)(i), we fix v ∈ V and G ∈ W. By the continuity of Wπ,ϕ , there exist u ∈ V and H ∈ W such that Wπ,ϕ (N(u, H) ⊆ N(v, G). Let ω = 2u. Then we claim that ω(ϕ(x))y ∈ H implies that v(x)πx (y) ∈ G for all x ∈ X, y ∈ E. [Assume that for some xo ∈ X and yo ∈ E, we have ω(ϕ(xo ))yo ∈ H but / G. Let D = {x ∈ X : ω(x)yo ∈ H}. Then D is an open v(xo )πxo (yo) ∈ neighborhood of ϕ(xo ) and therefore there exists f ∈ CVo (X) such that 0 ≤ f ≤ 1, f (ϕ(xo )) = 1 and f (X\D) = 0. Define g = 2f ⊗ yo. Now, it is easy to see that g ∈ N(v, H) and therefore we have Wπ,ϕ g ∈ N(v, G). Thus it follows that v(xo )πxo (yo ) ∈ 21 G ⊆ G, which is a contradiction. This proves our claim.] To prove (b)(ii), let v ∈ V and G ∈ W and compact set K ⊆ X. There exists f ∈ CVo (X) such that f (K) = 1. Fix 0 6= y ∈ E and defineg = f ⊗ y. Then clearly Wπ,ϕ g ∈ CVo (X, E). Now if we put D = {x ∈ X : v(x)πx (g(ϕ(x))) ∈ / G} and S = ϕ−1 (K) ∩ {x ∈ X : v(x)πx (y) ∈ / G}, then clearly D is a compact subset of X such that S ⊆ D. Thus S, being a closed subset of D, is compact. (b) ⇒ (a) Suppose that conditions (b)(i) and (b)(ii) hold. First of all we show that Wπ,ϕ is an into map. [Let f ∈ CVo (X, E) and let K ⊆ X be a compact set. Let {xα : α ∈ I} be a net in K such that xα → x. Fix G ∈ W. Choose balanced H ∈ W such that H + H ⊆ G. Let B = {f (ϕ(xα )) : α ∈ I}, a bounded set in E. Since π : X → CLu (E) is continuous and U(B, H) is a neighborhood of 0 in CLu (E), there exists αo ∈ I such that (πxα − πx )(f (ϕ(xα ))) ∈ H for all α ≥ αo . Again, since πx (f (ϕ(x))) → πx (f (ϕ(x))) in E, there exists αo ∈ I such that π(x)(f (ϕ(xα ))) − πx (f (ϕ(x))) ∈ H for all α ≥ αo . Let α1 ∈ I be such that α1 ≥ αo and α1 ≥ αo . Then, for every α ≥ α1 , we have πxα (f (ϕ(xo ))) − πx (f (ϕ(x)))

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= πxα (f (ϕ(xo ))) − πx (f (ϕ(x))) + πx (f (ϕ(xα )) − πx (f (ϕ(x))) ∈ H + H ⊆ G. This proves that Wπ,ϕ f ∈ C(X, E). To show that Wπ,ϕ f ∈ CVo (X, E), fix v ∈ V and G ∈ W. We need to prove that the set S = {x ∈ X : v(x)πx (f (x))) ∈ / G} is compact. Choose a balanced G1 ∈ W with G1 + G1 ⊆ G. According to condition (b)(i), there exist u ∈ V and H ∈ W such that u(ϕ(x))y ∈ H implies that v(x)πx (y) ∈ G1 for allx ∈ X, y ∈ E. Now, if we put K = {x ∈ X : v(x)f (x) ∈ / H}, then K is a compact subset of X and ϕ(S) ⊆ K. Let m = sup{u(x) : x ∈ K}. Obviously m > 0. Since f (ϕ(S)) is precompact in E, there exist a finite set {f (ϕ(xi )) : 1 ≤ i ≤ n}in f (ϕ(S)) such that n [ 1 f (ϕ(S)) ⊆ {f (ϕ(xi )) + H}. 2m i=1 Again, condition (b)(ii) implies that for each i ∈ {1, 2, ..., n} the set Ci = ϕ−1 (K) ∩ {x ∈ X : v(x)πx (f (ϕ(xi ))) ∈ / G1 } is compact. Now S ⊆ ∪ni=1 Ci , as follows. [Let xo ∈ S. Then there exists j ∈ {1, 2, ..., n} such that 1 H. f (ϕ(xo )) − f (ϕ(xj )) ∈ 2m That is, 1 u(ϕ(xo ))[f (ϕ(xo )) − f (ϕ(xj ))] ∈ H. 2m Further, it implies that v(xo )πxo [f (ϕ(xo )) − f (ϕ(xj ))] ∈ G1 . Now, we claim that / G1 . v(xo )πxo f (ϕ(xj ))) ∈ Assume the opposite. Then v(xo )πxo (f (ϕ(xo ))) = v(xo )πxo [f (ϕ(xo )) − f (ϕ(xj ))] + v(xo )πxo f (ϕ(xj )) ∈ G1 + G1 ⊆ G which is impossible. Thus xo ∈ Cj ⊆ ∪ni=1 Ci .] Therefore S, being a closed subset of the compact set ∪ni=1 Ci , is itself compact. This proves that Wπ,ϕ f ∈ CVo (X, E). To show that Wπ,ϕ is continuous, fix v ∈ V and G ∈ W. Then, by condition (i), there exist u ∈ V and H ∈ W such that u(ϕ(x))y ∈ H


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implies that v(x)πx (y) ∈ G for all x ∈ X and y ∈ E. We claim that Wπ,ϕ (N(u, H)) ⊆ N(v, G). [Let f ∈ CVo (X, E) be such that f ∈ N(u, H). Then we have u(ϕ(x))f (ϕ(x)) ∈ H for all x ∈ X. Further, it implies that v(x)πx (f (ϕ(x))) ∈ G for all x ∈ X. That is Wπ,ϕ f ∈ N(v, G).] This proves that Wπ,ϕ is continuous at the origin in CVo (X, E) and hence continuous on CVo (X, E). Thus Wπ,ϕ is a weighted composition operator on CVo (X, E). In the next theorem, we shall characterize the weighted composition operators Wπ,ϕ on CVo (X, E) induced by the mappings π : X → E and ϕ : X → X, where E is a topological algebra. Theorem 9.2.2. [MS98] Let X be a completely regular kR -space, and let E be a topological algebra with hypocontinuous multiplication and identity e. Let ϕ : X → X be a continuous map. Then the following conditions are equivalent: (a) Wπ,ϕ is a weighted composition operator on CVo (X, E). (b) (i) π : X → E is continuous; (ii) for any v ∈ V and G ∈ W there exist u ∈ V and H ∈ W such that u(ϕ(x))y ∈ H implies that v(x)πx (y) ∈ G for all x ∈ X, y ∈ E; (iii) for any v ∈ V , G ∈ W and compact set K ⊆ X, the set ϕ−1 (K) ∩ {x ∈ X : v(x)πx (y) ∈ / G} is compact for each y(6= 0) ∈ E. Proof. (a) ⇒ (b) If Wπ,ϕ is a weighted composition operator on CVo (X, E), then we need to prove condition (b)(i), whereas the proof of conditions (b)(ii) and (b)(iii) can be obtained with slight modifications as in the proof of conditions (b)(i) and (b)(ii) of Theorem 9.2.1. Fix xo ∈ X. Then there exists f ∈ CVo (X, E) such that f (ϕ(xo )) 6= 0. Also, there exists an open neighborhood D of xo in X such that f (ϕ(x)) 6= 0 for all x ∈ D. Let h : D → C be defined as h(x) = (f (ϕ(x)))−1 for all x ∈ D. Let g : X → E be defined as g(x) = f (x)e for all x ∈ X. Then clearly g ∈ CVo (X, E) and therefore Wπ,ϕ g ∈ CVo (X, E). Now, since the mapping π · g ◦ ϕ · h is continuous at xo , it follows that the mapping π is continuous at xo . This proves that the mapping π is continuous on X. (b) ⇒ (a) Suppose that conditions (b)(i) through (b)(iii) hold. Let f ∈ CVo (X, E) and let K ⊆ X be a compact set. Fix xo ∈ K and G ∈ W. Then there exists H ∈ W such that H + H ⊆ G. Let B1 = f (ϕ(K)) and B2 = {π(xo )}. Then B1 , B2 ∈ b(E). Since the multiplication in E is hypocontinuous, there exist H1 , H2 ∈ W such that H1 B1 ⊆ H and H2 B2 ⊆ H. Since the map π is continuous at xo , there exists an open

156


neighborhood D1 of xo in K such that π(x) − π(xo ) ∈ H1 for all x ∈ D1 . Again, since f ◦ ϕ is continuous at xo , there exists an open neighborhood D2 of xo in K such that f (ϕ(x)) − f (ϕ(xo )) ∈ H2 for all x ∈ D2 . f (ϕ(x)) − f (ϕ(xo )) ∈ H2 , for all x ∈ D2 . Let D = D1 ∩ D2 . Then, for any x ∈ D, we have π(x)f (ϕ)) − π(xo )f (ϕ(xo )) = (π(x) − π(xo )f (ϕ(x)) + π(xo )(f (ϕ(x)) − f (ϕ(xo ))) ∈ H1 B1 + H2 B2 ⊆ H + H ⊆ G. This proves that π · f ◦ ϕ is continuous at xo in K and hence on X. Thus Wπ,ϕ f ∈ C(X, E). Again with slight modifications in the proof of Theorem 9.2.1, it can be shown that Wπ,ϕ f ∈ CVo (X, E) and Wπ,ϕ is continuous on CVo (X, E). Remark 9.2.3. In case X is a completely regular kR -space and E is a topological algebra with hypocontinuous multiplication not necessarily containing an identity e, then the continuous mappings π : X → E and ϕ : X → X induce the weighted composition operator Wπ,ϕ on CVo (X, E) iff conditions (b)(ii) and (b)(iii) of Theorem 9.2.2. are satisfied. Theorem 9.2.4. [MS98] Let X be a completely regular space and E be a TVS, and let ϕ : X → X be a mapping. Then the following are equivalent: (a) ϕ induces the composition operator Cϕ on CVo (X, E); (b) ϕ induces the composition operator Cϕ on CVo (X, E) and CVo (X, E) is invariant under Cϕ ; (c) (i) ϕ is continuous, (ii) for any v ∈ V and G ∈ W there exist u ∈ V and H ∈ W such that u(ϕ(x))y ∈ H implies that v(x)y ∈ G for all x ∈ X, y ∈ E, (iii) for any v ∈ V , G ∈ W and compact set K ⊆ X, the set ϕ−1 (K) ∩ {x ∈ X : v(x)y ∈ / G} is compact for each y(6= 0) ∈ E. Proof. (a) ⇒ (c) In view of assumption (∗∗), it follows that if ϕ : X → X is any function for which the corresponding composition map Cϕ induced on CVo (X, E) has its range contained in C(X, E), then ϕ is necessarily continuous. Hence (c)(i) holds.


157

To prove condition (c)(ii), we fix v ∈ V and G ∈ W. Since Cϕ is continuous on CVo (X, E), there exist u ∈ V and H ∈ W such that Cϕ (N(u, H)) ⊆ N(v, G). Let ω = 2u. We claim that ω(ϕ(x))y ∈ H implies that v(x)y ∈ G for all x ∈ X and y ∈ E. Assume that for some xo ∈ X and yo ∈ E, we have ω(ϕ(xo))yo ∈ H but v(xo )yo ∈ / G. Let D = {x ∈ X : ω(x)yo ∈ H}. Then D is an open neighborhood of ϕ(xo ) and therefore there exists f ∈ CVo (X) such that 0 ≤ f ≤ 1, f (ϕ(xo )) = 1 and f (X\D) = 0. Define g = 2f ⊗ yo . Then g ∈ N(u, H) and thereby Cϕ g ∈ N(v, G). Further, it implies that v(xo )yo ∈ G, which is impossible. Thus condition c(ii) is satisfied. To see condition (c)(iii), let v ∈ V and G ∈ W and compact set K ⊆ X. Then there exists f ∈ CVo (X) such that f (K) = 1. Fix yo(6= 0) ∈ E. Define g = f ⊗ yo . Let S = {x ∈ X : v(x)g(ϕ(x)) ∈ / G} and M = ϕ−1 (K) ∩ {x ∈ X : u(x)yo ∈ / G}. Then obviously S is a compact subset of X and M ⊆ S. Thus M is compact, being a closed subset of S. This proves condition (c). (c) ⇒ (b) To show that Cϕ is a composition operator on CVb (X, E), we fix v ∈ V and G ∈ W. By condition (c)(ii), there exist u ∈ V and H ∈ W such that u(ϕ(x))y ∈ H implies that v(x)y ∈ G1 , for all x ∈ X and y ∈ E. Now we claim that Cϕ (N(u, H)) ⊆ N(v, G). [Let f ∈ N(u, H). Then since u(ϕ(x))f (ϕ(x)) ∈ H for all x ∈ X, it follows that v(x))f (ϕ(x)) ∈ G for all x ∈ X. That is, Cϕ f ∈ N(v, G). This proves the continuity of Cϕ at the origin in CVb (X, E) and hence Cϕ is continuous on CVb (X, E). To complete the proof of condition (b), it remains to show that CVo (X, E) is invariant under Cϕ . [Fix f ∈ CVo (X, E). Let v ∈ V and G ∈ W. Consider the set S = {x ∈ X : v(x)f (ϕ(x)) ∈ / G}. Choose a balanced G1 ∈ W with G1 + G1 ⊆ G. According to condition (c)(ii), there exist u ∈ V and H ∈ W such that u(ϕ(x))y ∈ H implies that v(x)y ∈ G1 for all x ∈ X, y ∈ E. Let K = {x ∈ X : u(x)f (x) ∈ / H}. Then K is a compact subset of X and ϕ(S) ⊆ K. Let λ = sup{u(x) : x ∈ K}. Then obviously λ > 0. Since f (ϕ(S)) is precompact in E, there exists a finite set {f (ϕ(xi ))}ni=1

158


in f (ϕ(S)) such that f (ϕ(S)) ⊆

n [

{f (ϕ(xi )) +

i=1

1 H}. 2λ

In view of condition (c)(iii), it follows that for each i ∈ {1, 2, ..., n}, the set Ai = ϕ−1 (K) ∩ {x ∈ X : v(x)f (ϕ(xi )) ∈ / G1 } n is compact. If we put A = ∪i=1 Ai , then A is a compact subset of X. Now, to show that S ⊆ A, let y ∈ S. Then there exists j ∈ 1 {1, 2, ..., n} such that f (ϕ(y)) − f (ϕ(xj )) ∈ 2λ H. Further, it implies 1 that u(ϕ(y))[f (ϕ(y)) − f (ϕ(xi ))] ∈ 2 H. By hypothesis, it follows that v(y)[f (ϕ(y)) − f (ϕ(xj ))] ∈ G1 . Now, if v(y)(f (ϕ(xi )) ∈ G1 , v(y)f (ϕ(y)) = v(y)[f (ϕ(y)) − f (ϕ(xj ))] + v(y)(f (ϕ(xi )) ∈ G1 + G1 ⊆ G, a contradiction (since y ∈ S). Hence v(y)(f (ϕ(xi )) ∈ / G1 ,showing that y ∈ A. Thus S, being a closed subset of A, is compact. This proves that Cϕ f ∈ CVo (X, E).] (b) ⇒ (a). This is clear. Remark 9.2.5. If ϕ : X → X is a surjective homeomorphism, then condition (c)(iii) is automatically satisfied and such a function will therefore induce a composition operator on CVo (X, E) as soon as condition (c)(ii) is satisfied.

3. COMPACT WEIGHTED COMPOSITION OPERATORS

159

3. Compact Weighted Composition Operators This section contains characterization of compact weighted composition operators and, in particular, of precompact, equicontinuous and bounded multiplication operators on the weighted space CVo (X, E), where E is a non-trivial TVS. As before, it is assumed that the Nachbin family V satisfies (∗) V > 0 and (∗∗) CVo (X) does not vanish on X. Theorem 9.3.1. [MS98] Let X be a locally compact space and E a TVS. Let V ⊆ So+ (X) and π : X → CLu (E) and ϕ : X → X be mappings with ϕ continuous, and suppose that Wπ,ϕ is a weighted composition operator on CVo (X, E) Then the following conditions are equivalent: (a)Wπ,ϕ is a compact operator on CVo (X, E). (b)(1) π : X → CLu (E) is continuous. (2) For every x ∈ X, πx := π(x) is a compact operator on E. (3) ϕ is locally constant on X\z(π), where z(π) = {x ∈ X : π(x) = 0}. Proof. (a) ⇒ (b)) Suppose Wπ,ϕ is a compact operator on CVo (X, E). Then conditions (b)(1)-(3) hold, as follows. (b) (1) Let xo ∈ X, B ∈ b(E), G ∈ W. Since X is locally compact, choose a neighborhood Ko of ϕ(xo ) such that cl-Ko is compact. Choose f ∈ CVo (X) such that f (cl-Ko) = 1. Let Do = ϕ−1 (Ko ). For each y ∈ B, define gy = f ⊗ y. Then the set M = {gy : y ∈ B} is bounded in CVo (X, E). Further, it implies that the set Wπ,ϕ (M) is relatively compact in CVo (X, E). Now, by the Arzela-Ascoli Theorem 3.1.4, the set Wπ,ϕ (M) is equicontinuous at xo . Thus there exists a neighborhood D1 of xo such that πx (gy (ϕ(x))) − πxo (gy (ϕ(xo ))) ∈ G for all x ∈ D1 , y ∈ B. Let D = Do ∩ D1 . Then we have πx (y) − πxo (y) ∈ G for all x ∈ D, y ∈ B. That is, πx − πxo ∈ N(B, G) for all x ∈ D. This proves that π : X → CLu (E) is continuous at xo and hence on X. (b)(2) Let xo ∈ X, B ∈ b(E). We need to show that πxo (B) is relatively compact in E. Choose f ∈ CVo (X) such that f (ϕ(xo ) = 1. For each y ∈ B, define gy = f ⊗ y. Let F = {gy : y ∈ B}. Then it follows that Wπ,ϕ (F ) is relatively compact in CVo (X, E). So, by Theorem 3.1.4,

160


the set Wπ,ϕ (F )(xo ) is relatively compact in E. But Wπ,ϕ (F )(xo ) = {πxo (gy (ϕ(xo ))) : y ∈ B} = = {πxo (f ((ϕ(xo )y) : y ∈ B} = πxo (B); hence πxo (B) is relatively compact in E. This proves that πxo is a compact operator on E. (b)(3) Suppose ϕ is not locally constant on X\z(π). Then there exists xo ∈ X\z(π) such that ϕ is not locally constant on any open neighborhood D of xo . Let B be any open base at xo . The set B can be ordered by the relation D1 ≤ D2 iff D1 ⊆ D2 . Thus there exists a net {xD : D ∈ B} in X such that xD → xo such that ϕ(xD ) 6= ϕ(xo ). For each D ∈ B, we choose fD ∈ Cb (X) such that fD (ϕ(xo ) = 1 and fD (ϕ(xD ) = 0. Again choose h ∈ CVo (X) such that h(ϕ(xo ) = 1. Let y ∈ E be such that / H. For each πxo (y) 6= 0. Then there exists H ∈ W such that πxo (y) ∈ D ∈ B, define gD = (hfD ) ⊗ y. Since CVo (X) is a Cb (X)-module, it easily follows that the set M = {gD : D ∈ B} is bounded in CVo (X, E) and consequently the set Wπ,ϕ (M) is relatively compact in CVo (X, E). Again, by Theorem 3.1.4, the set Wπ,ϕ (M) is equicontinuous at xo . Thus, for a given net xD → xo and H ∈ W, we have sup{ρH [πxD (gD (ϕ(xD ))) − πxo (gD (ϕ(xo )))] : D ∈ B} → 0. / H. This That is, ρH (πxo (y)) → 0, which is impossible because πxo (y) ∈ proves that ϕ is locally constant on X\z(π). (b) ⇒ (a) Suppose that the conditions (b)(1)-(3) hold. Let F ⊆ CVo (X, E) be a bounded set. We need to show that Wπ,ϕ (F ) is compact in CVo (X, E). It is enough to show that the set cl-Wπ,ϕ (F ) satisfies conditions (i) -(iii) of the Arzela-Ascoli Theorem 3.1.4. (i) Clearly, cl-Wπ,ϕ (F ) is β-closed. Since the closure of a bounded set is bounded, it follows that cl-Wπ,ϕ (F ) is β-bounded. (ii) Let xo ∈ X and B = {f (ϕ(xo )) : f ∈ F }. Then clearly the set B is bounded in E. Since πxo is a compact operator, it readily follows that cl-Wπ,ϕ (F )(xo ) = cl-{πxo (f (ϕ(xo ))) : f ∈ F } = cl-πxo (B), which is relatively compact in E. (iii) To show that cl-Wπ,ϕ (F ) is equicontinuous on each compact subset of X, it suffices that Wπ,ϕ (F ) is equicontinuous on each compact subset of X. Suppose K is any compact subset of X. Let xo ∈ K and G ∈ W. Consider the set S = {f (ϕ(x)) : x ∈ X, f ∈ F }. Then S is bounded in E. In case xo ∈ z(π), it is easy to see that Wπ,ϕ (F ) is equicontinuous at xo . Now suppose xo ∈ / z(π). Then, by condition (3),


161

there exists an open neighborhood D1 (xo ) of xo such that ϕ is constant on D1 (xo ). Since π : X → CLu (E) is continuous, there exists an open neighborhood D2 (xo ) of xo such that πx − πxo ∈ N(S, G) for all x ∈ D2 (xo ). Let D = D1 (xo ) ∩ D2 (xo ). Then D is a neighborhood of xo in K such that πx (f (ϕ(x))) − πxo (f (ϕ(xo ))) ∈ G for all x ∈ D and f ∈ F. So Wπ,ϕ (F ) is equicontinuous at xo and hence on K. Theorem 9.3.2. [MS98] Let X be a locally compact space and E a Hausdorff topological algebra with hypocontinuous multiplication containing an identity e. Let V ⊆ So+ (X) and π : X → CLu (E) and ϕ : X → X be mappings with ϕ continuous, and suppose that Wπ,ϕ is a weighted composition operator on CVo (X, E). Then the following conditions are equivalent: (a) Wπ,ϕ is a compact weighted composition operator on CVo (X, E). (b)(1) π : X → CLu (E) is continuous; (2) For every x ∈ X, πx := π(x) is a compact operator on E; (3) ϕ is locally constant on X\z(π). Proof. The proof can be obtained by slight modification in the proof of Theorem 9.2.2 and Theorem 9.3.1. Corollary 9.3.3. [MS98] Let X be a locally compact space and E a an infinite dimensional TVS, and let V ⊆ So+ (X). Then there is no compact composition operator on CVo (X, E). Proof. We first note if π : X → CLu (E) is the map given by π(x) = I for all x ∈ X, then the weighted composition operator Wπ,ϕ becomes the composition operator Cϕ . Since E is an infinite dimensional TVS, the identity operator I on E is not compact. Consequently, the condition (b) of Theorem 9.3.2 cannot hold. Remarks. In Theorems 9.3.1 and 9.3.2, if we take U as a system of weights on X such that U ⊆ V, then the results hold for the space CUo (X, E). (ii) Theorems 9.3.1 and 9.3.2 make sure that there are non-zero compact weighted composition operator on these spaces of continuous functions, where as it is not the case with Lp -spaces. However, there need not be non-zero compact weighted composition operator on Lp -spaces (with non-atomic measures) [Tak92]. Example. Let X = N with the discrete topology, and let E = Cb (R), Let π : N → CL(E) be defined by π(n) = n1 An for all n ∈ N, where

162


An : E → E is defined by An f (t) = f (n) for all f ∈ E and t ∈ R. Clearly, π(n) is a compact operator on R. Let ϕ : N → N be defined by ϕ(n) = n2 , n ∈ N. Let ω : N → R+ be defined by ω(n) = n1 , n ∈ N. Consider V = {λω : λ ≥ 0}. Now, it is easy to see that the mappings π and ϕ satisfy all the conditions of Theorem 9.3.1 and therefore Wπ,ϕ is a compact weighted composition operator on CVo (X, E). We next consider the particular case of multiplication operators but characterize more general notions of equicontinuous, precompact and bounded multiplication operator Mπ on CVo (X, E) and CVb (X, E). Recall that a linear map T : CVb (X, E) −→ CVb (X, E) is said to be compact (resp. precompact, equicontinuous, bounded ) if it maps some 0neighborhood in CVb (X, E) into a compact (resp. precompact, equicontinuous, bounded) subset of CVb (X, E). Lemma 9.3.4. [Oub02, AAK09] For any v ∈ V , G ∈ W and x ∈ X, 1 = sup{ρG (f (x)) : f ∈ N(v, G)} v(x) = sup{ρG (f (x)) : f ∈ CVb (X, E) with kf kv,G ≤ 1}. (1) Proof. Let x ∈ X, v ∈ V , G ∈ W. There exist f ∈ CVb (X, E) and H ∈ W such that (ρH ◦ f )(x) = 1. Choose a ∈ E with ρG (a) = 1. Case I. Suppose v(x) = 0. For each n ≥ 1, set 1 1 1 Un := {y ∈ X : v(y) < and 1 − < ρH (f (y)) < 1 + }; n n n and consider hn ∈ Cb (X) with 0 ≤ hn ≤ n, hn (x) = n, and supp(hn ) ⊆ Un . Clearly, the function gn :=

n h n+1 n

ρH ◦ f ⊗ a ∈ CVb (X, E) and

n hn (y)ρH (f (y))ρG (a) : y ∈ X} n+1 n = sup{v(y) hn (y)ρH (f (y))ρG (a) : y ∈ Un } n+1 1 n 1 < · · n · (1 + ) · 1 = 1, n n+1 n

kgn kv,G = sup{v(y)

hence, sup{ρG (f (x))

f ∈ CVb (X, E), kf kv,G ≤ 1} ≥ sup{ρG (gn (x)) : n ∈ N} n hn (x)ρH (f (x))ρG (a) : n ∈ N} = sup{ n+1 n 1 = sup{ · n · 1 · 1 : n ∈ N} = ∞ = . n+1 v(x) :


Case II. Suppose v(x) 6= 0. For n >

1 , v(x)

163

let

v(x) v(x) 1 ≤ ρH (f (y)) ≤ 1 }, v(x) + 2n v(x) − 2n v(x) v(x) }. : = {y ∈ X : 1 < ρH (f (y)) < v(x) + n v(x) − n1

Fn : = {y ∈ X : Un

Then choose hn , kn ∈ Cb (X) with 1 1 0 ≤ hn ≤ 1 , hn (x) = v(x) + n v(x) +

1 n

on Fn , and supp(hn ) ⊆ Un ,

0 ≤ kn ≤ 1, kn (x) = 1, and supp(kn ) ⊆ Jn := Fn ∩ {y ∈ X : v(y) < v(x) + The function gn :=

1 v(x)− 2n v(x)

1 }. n

hn kn ρH ◦ f ⊗ a ∈ CVb (X, E) and

kgn kv,G = sup{v(y)ρG (gn (y)) : y ∈ X} 1 v(x) − 2n hn (y)kn (y)ρH (f (y))ρG (a) : y ∈ X} = sup{v(y) v(x) 1 v(x) − 2n = sup{v(y) hn (y)kn (y)ρH (f (y))ρG (a) : y ∈ Jn } v(x) 1 v(x) 1 v(x) − 2n 1 < (v(x) + ) · 1 ·1 1 = 1; 2n v(x) v(x) + 2n v(x) − 2n

hence sup{ρG (f (x))

:

f ∈ CVb (X, E), kf kv,G ≤ 1}

≥ sup{ρG (gn (x)) : n ∈ N} 1 v(x) − 2n 1 1 = sup{ · . 1 · 1 · 1 · 1 : n ∈ N} = v(x) v(x) v(x) − 2n On the other hand, sup{ρG (f (x))

f ∈ CVb (X, E), kf kv,G ≤ 1} 1 = sup{v(x)ρG (f (x)) : f ∈ CVb (X, E), kf kv,G ≤ 1} v(x) 1 sup{supv(y)ρG (f (y)) : f ∈ CVb (X, E), kf kv,G ≤ 1} ≤ v(x) y∈X :

164


=

1 1 sup{kf kv,G : f ∈ CVb (X, E), kf kv,G ≤ 1} = . v(x) v(x)

Thus, in each case, (1) holds. Recall that if A ⊆ X, then a point x ∈ A is called an isolated point of A if x is not a limit point of A, i.e. if there exists a neighborhood U of x in X such that U ∩ A = {x}. Consequently, if X has no isolated point, then, for each x ∈ X, Theorem 9.3.5. [Oub02, AAK09] Let π : X −→ CL(E) be a map such that Mπ (CVb (X, E)) ⊆ C(X, E) and supposeX has no isolated points. If Mπ is equicontinuous on CVb (X, E), then Mπ = 0. Proof. Suppose Mπ is equicontinuous on CVb (X, E) but Mπ (fo ) 6= 0 for some fo ∈ CVb (X, E). Then there exists xo ∈ X with πxo (fo (xo )) 6= 0. Since Mπ is equicontinuous, there exist some v ∈ V and balanced G ∈ W such that Mπ (N(v, G)) is equicontinuous on X and in particular at xo . We may assume that fo ∈ N(v, G) (since N(v, G) is absorbing). Hence, for every balanced H ∈ W, there exists a neighborhood D of xo in X such that πy (f (y)) − πxo (f (xo )) ∈ H for all y ∈ D and f ∈ N(v, G). Since xo is not isolated, there exists some y ∈ D with y 6= xo . Choose then gy ∈ Cb (X) with 0 ≤ gy ≤ 1, gy (y) = 0, and gy (xo ) = 1. Then gy fo ∈ N(v, G) and so πy (gy (y)fo(y)) − πxo (gy (xo )fo (xo )) ∈ H; that is, πxo (fo (xo )) ∈ H. Since H ∈ W is arbitrary and E is Hausdorff (i.e. ∩ U = {0}), we have πxo (fo (xo )) = 0. This is the desired U ∈W

contradiction.

Corollary 9.3.6. [Oub02, AAK09] Let π : X −→ CL(E) be a map such that Mπ (CVb (X, E)) ⊆ C(X, E). If X is a VR -space without isolated points, then Mπ is precompact iff Mπ = 0. Proof. Suppose Mπ is precompact. Then, by Theorem 3.2.1, Mπ is equicontinuous. Hence, by Theorem 9.3.5, Mπ = 0. The converse is trivial. Theorem 9.3.7. [Oub02, AAK09] Let π : X −→ CL(E) be such that Mπ (CVb (X, E)) ⊆ C(X, E). Then Mπ is a bounded multiplication operator on CVb (X, E) iff there exist v ∈ V and G ∈ W such that for any u ∈ V, H ∈ W, there exists λ > 0 such that λu(x)a ∈ G implies that u(x)πx (a) ∈ H, x ∈ X, a ∈ E,


165

or equivalently u(x)ρH (πx (a)) ≤ λv(x)ρG (a), x ∈ X, a ∈ E. Proof. Similar to that of Theorem 9.1.1.

(A)

166


4. Notes and Comments Section 9.1. Singh and Summers [SS88] have studied the notion of composition operators on CVo (X, C). Later, Singh and Manhas [SM91, SM92] made an analogous study of multiplication operators Mθ , Mψ and Mπ from CVb (X, E) (resp. CVo (X, E)) into itself, assuming E a locally convex space or a locally m-convex algebra. These operators are induced by certain maps θ : X → C and ψ : X → E and π : X → CL(E).This section contains results from the papers [KT97, KT02, MS98] which generalize the results of above authors to the case when E is a TVS or topological algebra which is not necessarily locally convex. (See [SK97, Oub05]). Section 9.2. This section contains characterizations of the more general notion of weighted composition operators Wπ,ϕ from CVb (X, E) (resp. CVo (X, E)) into itself, due to Manhas and Singh [MS98]. Here ϕ : X → X and π : X → CL(E). Section 9.3. A systematic study of compact weighted composition operators on spaces of continuous functions was initiated by Kamowitz [Kam81] on the Banach algebra C(X) for a compact Hausdorff space X. This study has been further extended to function algebra setting and Banach lattices by Feldman [Fel90], Singh and Summers [SS87] and Takagi [Tak88]. Jamison and Rajagopalan [JR88] and Takagi [Tak91] have further generalized these results to the spaces C(X, E). Finally, in [MS97] Manhas and Singh have obtained a study of compact and weakly compact weighted composition operators on the weighted locally convex spaces CVo (X, E), where V ⊆ So+ (X) is a Nachbin family on X and E is a quasi-complete locally convex space. This section contains characterization of compact Weighted Composition Operators Wπ,ϕ and also of precompact, equicontinuos and bounded multiplication operators on the non-locally convex weighted spaces CVo (X, E), as given in [MS98, Oub02, AAK09] for E a non-trivial TVS. More general results for operators Wπ,ϕ from CVb (X, E) into CUb (Y, E) can be found in [Oub05]; here X and Y are completely regular spaces and ϕ : Y → X and π : Y → CL(E) are the corresponding maps maps.

CHAPTER 10

The General Strict Topology The general strict topology was defined by Busby in [Bus68] in the case of Y a C ∗ -algebra and A a closed two-sided ideal in Y as the one given by the seminorms x → max{kaxk, kxak} for a ∈ A, x ∈ Y. Independently, this topology was studied in greater detail and in a more general setting of Y a Banach left A-module by Sentilles and Taylor in an interesting paper [ST69] as the one given by the seminorms x → ka.xk for a ∈ A, x ∈ Y . More recently, Khan-Mohammad-Thaheem [KMT05] have considered the case of Y a locally idempotent F -algebra which contains A as a closed two-sided ideal, while Shantha [Shan04, Shan05] has taken Y a locally convex module over a locally convex algebra A. In this chapter, we begin with the study of general strict and related topologies on a topological left A-module Y, where Y is Hausdorff TVS and A a Hausdorff topological algebra having a two-sided bounded approximate identity. These results were originally given in the papers [ST69, Shan04, KMT05, SumW72] in particular cases and extended recently in [Kh07, Kh08]. We shall also consider essentiality of topological left A-modules (Cb (S, E), u) and (C(S, E), k). Finally, we study the properties of the topological module HomA (A, Y ) of continuous homomorphisms.

167

168

10. THE GENERAL STRICT TOPOLOGY

1. Strict Topology on Topological Modules Recall that if Y is a TVS and A be a topological algebra, both over the same field K (= R or C), then Y is called a topological left A-module if it is a left A-module and the module multiplication (a, x) → a.x from A × Y into Y is separately continuous. If b(A) (resp. b(Y )) denote the collection of all bounded sets in A (resp. Y ), then module multiplication (a, x) → ax is called b(A)- (resp. b(Y )-) hypocontinuous if, given any neighborhood G of 0 in Y and any D ∈ b(A) (resp. B ∈ b(Y )), there exists a neighborhood H of 0 in Y (resp. V of 0 in A) such that D.H ⊆ G (resp. V.B ⊆ G). We mention that if (Y, τ ) is an ultrabarrelled topological left A-module, then the module multiplication is b(A)-hypocontinuous. (See section A.6). Definition. Let A be a topological algebra and (Y, τ ) a TVS which is a topological left A-module, and let W be a base of τ neighborhoods of 0 in Y . For any bounded set D ⊆ A and G ∈ W, we set M(D, G) = {x ∈ Y : D.x ⊆ G}. The uniform topology u = uA (resp. strict topology β = βA ) on Y is defined as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form M(D, G), where D is a bounded (resp. finite) subset of A and G ∈ W. If A has a two-sided approximate identity {eλ : λ ∈ I}, the compact-open topology k on Y is defined just as above with D being a finite subset of {eλ : λ ∈ I}. As mentioned earlier, β was introduced by Sentilles and Taylor ([ST69], p.145) in the case of Y a left Banach A-module as the one given by the seminorms x → ka.xk for a ∈ A, x ∈ Y . As a classical example, if A = Co (S) and Y = Cb (S, E) with S a locally compact space and E a TVS and Y endowed with τ = u, the uniform topology and A endowed with the sup-norm topology and has an approximate identity. In this particular case, β is the strict topology and k is the topology of uniform convergence on compact subsets of S. Another instance of the above is that A is a topological algebra and Y = CMd (A), the set of all double multipliers (S, T ), where A is a topological algebra and S, T : A → A are continuous linear mappings satisfying aS(b) = T (a)b for all a, b ∈ A [Jo64, KMT99]. We shall consider more cases of essential topological left A-modules (Cb (S, E), u) and (C(S, E), k) in the later part of this section. In the sequel, we shall assume, unless stated otherwise, that Y is a topological left A-module, where Y = (Y, τ ) is a Hausdorff TVS with

1. STRICT TOPOLOGY ON TOPOLOGICAL MODULES

169

W a base of neighborhoods of 0 and A a Hausdorff topological algebra with {eλ : λ ∈ I} a bounded two-sided approximate identity. We shall also assume that A is faithful in Y in the sense that, for any x 6= 0 in Y , a.x = 0 for all a ∈ A implies that x = 0 (cf. [Jo64, ST69]). In some results, the assumption of ultrabarrelledness on A or Y is required essentially to apply the uniform boundedness principle. Theorem 10.1.1. [Kh07] Let (Y, τ ) be a topological left A-module with A having a two-sided approximate identity {eλ : λ ∈ I}.Then: (a) k 6 β 6 u. (b) k, β, and u are Hausdorff. β

(c) For each x ∈ Y , eλ .x −→ x. (d) ∪ eλ .Y is β-dense in Y ; in particular, Ye is β-dense in Y. λ∈I

If, in addition the module multiplication is b(A)-hypocontinuous, then: (e) u 6 τ. (f ) (Y, k), (Y, β) and (Y, u) are topological A-modules. Proof. (a) Since {eλ : λ ∈ I} ⊆ A, it follows that k 6 β. Since every finite set in A is bounded, we have β 6 u. (b) We only need to show that k is Hausdorff. Let x ∈ ∩ ∩ λ∈J G∈W

M(eλ , G). Since τ is Hausdorff,

∩ G = {0} and so eλ .x = 0 for all

G∈W

λ ∈ I. Then, for any a ∈ A, aeλ → a in A and so, by separate continuity of module multiplication, a.x = lim(aeλ ).x = lim a.(eλ .x) = 0. λ

λ

Since A is faithful in Y , we have x = 0. Thus (Y, k) is Hausdorff. (c) Let x ∈ Y , and let D be a finite set in A and G ∈ W. Since aeλ → a in A for all a ∈ D and D is finite, by separate continuity of module multiplication, there exists λo ∈ I such that for all λ ≥ λo and a∈D (aeλ − a).x ∈ G,

or a.(eλ .x − x) ∈ G. β

Hence eλ .x − x ∈ M(D, G) for all λ ≥ λo . So eλ .x −→ x. β

(d) Let x ∈ Y . Since {eλ .x : λ ∈ I} ⊆ ∪ eλ .Y and, by (c), eλ .x −→ x, λ∈I

we have x ∈ β-cl (∪ eλ Y ). Thus ∪ eλ .Y is β-dense in Y . λ∈I

λ∈I

τ

(e) We need to verify that, for any net {xα : α ∈ J} in Y , xα −→ 0 u implies that xα −→ 0. Let D be a bounded set in A and G ∈ W. By b(A)-hypocontinuity, choose balanced H ∈ W such that D.H ⊆ G. Since

170

10. THE GENERAL STRICT TOPOLOGY τ

D is bounded, there exists r > 1 such that D ⊆ rV . Since xα −→ 0, there exists αo ∈ J such that xα ∈ H for all α ≥ αo . Then, for any a ∈ D and α ≥ αo , a.xα ∈ D.H ⊆ G; u

that is, xα ∈ M(D, G) for all α ≥ αo . Hence xα −→ 0, and so u 6 τ . (f) We prove the result only for (Y, u) as the proofs for (Y, β) and (Y, k) are similar to it. First, let {bα : α ∈ J} ⊆ A with bα → 0 and fix u x(6= 0) in Y . To show that bα x −→ 0 in Y , let D be a bounded subset of A and G ∈ W. By b(A)-hypocontinuity, choose balanced H ∈ W such that D.H ⊆ G. Since bα → 0 in A, by separate continuity of module τ multiplication, bα .x −→ 0 in Y and so there exists αo ∈ J such that bα .x ∈ H for all α ≥ αo . Hence, for any a ∈ D and α ≥ αo , abα .x ∈ D.H ⊆ G; that is, bα .x ∈ u M(D, G), for all α ≥ αo . Thus bα .x −→ 0. u Next, let {xα : α ∈ J} ⊆ Y with xα −→ 0 and fix a(6= 0) in A. Let D be a bounded set in A and G ∈ W. Since the map Ra : A → A given by Ra (b) = ba, b ∈ A, is linear and continuous (by separate continuity of u multiplication in A), it follows that Da is bounded in A. Since xα −→ 0, there exists αo ∈ I such that for all α ≥ αo xα ∈ M(Da, G), or a.xα ∈ M(D, G). u

Hence a.xα −→ 0 in Y . Consequently, (Y, u) is a left semitopological A-module. Theorem 10.1.2. [Kh07] Let (Y, τ ) be a topological left A-module with b(Y )-hypocontinuous module multiplication and A having a twosided approximate identity {eλ : λ ∈ I}.Then: (p) k = β on τ -bounded sets. (q) If a sequence {xn } ⊆ Y is τ -bounded and k-convergent, then it is β-convergent. Proof. (p) Let S be a τ -bounded in Y , and let {xα : α ∈ J} be a net k in S with xα −→ x ∈ S. Let D be a finite set in A and G ∈ W. Choose balanced H ∈ W with H + H + H ⊆ G. By b(Y )-hypocontinuity, there exists a neighborhood V of 0 in A such that V.S ⊆ H. Since D is finite, there exists λo ∈ I such that


171

aeλo − a ∈ V for all a ∈ D. k

τ

τ

Since xα → xo , eλo .xα → eλo .xo and so, for each a ∈ D, a.(eλo .xα ) → a.(eλo .xo ). Since D is finite, choose αo ∈ J such that a.(eλo .xα − eλo .xo ) ∈ H for all α ≥ αo and a ∈ D. Then, for any a ∈ D and α ≥ αo , a.(xα − x) = (a − aeλo ).xα + aeλo .(xα − x)] + (aeλo − a).x ∈ V.S + H + V.S ⊆ H + H + H ⊆ G; β

that is, xα − x ∈ M(D, G) for all α ≥ αo . Hence xα −→ x, and so β 6 k on S. Thus k = β on S. k (q) Suppose that {xn } is τ -bounded and xn −→ x ∈ Y . By (1), k = β β

on the τ -bounded set {x, xn : n ≥ 1}; hence xn −→ x. Lemma 10.1.3. [Kh07] Let Y be a topological left A-module with A ultrabarralled. Let S be a β-bounded set in Y . Then, for any bounded set D ⊆ A, D.S is τ -bounded in Y . Proof. For any x ∈ Y , define Rx : A → Y by Rx (a) = a.x, a ∈ A. Clearly, each Rx is linear and also continuous (by separate continuity of module multiplication). We next show that {Rx : x ∈ S} is pointwise bounded in CL(A, Y ). Let a ∈ A and G ∈ W. Since S in β-bounded, there exists r > 0 such that S ⊆ rM(a, G). So {Rx (a) : x ∈ S} = {a.x : x ∈ S} ⊆ rG, showing that {Rx : x ∈ S} is pointwise bounded. Since A is ultrabarralled, by the principle of uniform boundedness, {Rx : x ∈ S} is equicontinuous and hence uniformly bounded in CL(A, Y ). Since D is bounded in A, D.S = ∪ Rx (D) is τ -bounded in Y . x∈S

Theorem 10.1.4. [Kh07] Let (Y, τ ) be a topological left A-module with A ultrabarralled. Then: (i) β and u have the same bounded sets in Y . (ii) If (Y, β) is ultrabornological, then β = u on Y . If, in addition, A has a two-sided approximate identity {eλ : λ ∈ I}, then: (iii) k = β on u-bounded sets. (iv) A sequence {xn } ⊆ Y is β-convergent iff it is u-bounded and k-convergent.

172


Proof. (i) Since β 6 u, every u-bounded set is β-bounded. Now, let S be any β-bounded set in Y . By Lemma 10.1.3, for any bounded set D ⊆ A, D.S is τ -bounded in Y or, equivalently, S is u-bounded. (ii) By (i), the identity map i : (Y, β) → (Y, u) takes bounded sets into bounded sets. Since Y is ultrabornological, i is continuous. Hence u 6 β. (iii) Let S be a u-bounded set in Y , and let {xα : α ∈ J} ⊆ S with k xα → xo ∈ S. Let D be a finite set in A and G ∈ W. Choose a balanced H ∈ W with H + H + H ⊆ G. As in the proof of Lemma 10.1.3, by the principle of uniform boundedness, {Rx : x ∈ S} is equicontinuous in CL(A, Y ). Hence there exists a neighborhood V of 0 in A such that V.S ⊆ H. Choose λo ∈ I such that aeλo − a ∈ V for all a ∈ D. k

τ

Since xα → xo , eλo .xα → eλo .xo and so by separate continuity, for each a ∈ D, τ aeλo .xα → aeλo .xo . Since D is finite, choose αo ∈ J such that aeλo .(xα − xo ) ∈ H for all α ≥ αo and a ∈ D. Hence, for any a ∈ D and α ≥ αo , a.(xα − xo ) = (a − aeλo ).xα + aeλo .(xα − xo ) + (aeλo − a).xo ∈ V.S + H + V.S ⊆ H + H + H ⊆ G. β

So xα → xo and hence β 6 k on S. β (iv) Let {xn } ⊆ Y with xn −→ x ∈ Y . Then it is β-bounded and k hence, by part (i), u-bounded. Further, since k 6 β, xn −→ x. Conk versely, suppose that {xn } is u-bounded and xn −→ x ∈ Y . By (iii), β

k = β on the u-bounded set {x, xn : n ≥ 1}; hence xn −→ x. The next result characterizes the equivalence of bounded sets in the topologies β, u, and τ on Y . Theorem 10.1.5. [Kh07] Let (Y, τ ) be a topological left A-module with A ultrabarralled. Then the following are equivalent: (a) u and τ have the same bounded sets in Y . (b) β and τ have the same bounded sets in Y . If the module multiplication is either b(A)-hypocontinuous or b(Y ) hypocontinuous, then each of (a) and (b) is equivalent to: (c) The β-closure of a τ -bounded set is τ -bounded.


173

Proof. (a) ⇔ (b) since β and u have the same bounded sets by Theorem 10.1.4(i). (b) ⇒ (c). Let S be a τ -bounded set in Y , and let D = {eλ : λ ∈ I}. Then D.S is τ -bounded in Y (cf. [Mal86], p. 28). [Indeed, let G ∈ W. By b(Y )-hypocontinuity, there exists a neighborhood V of 0 in A such that V.S ⊆ G. Since D is a bounded set in A, there exists r > 0 such that D ⊆ rV. Hence D.S ⊆ rV.S ⊆ rG.] Further, by b(A)-hypocontinuity and Theorem 10.1.1((a), (e)), β 6 τ and so D.S and hence β-cl(D.S) is ββ

bounded. By (b), β-cl(D.S) is τ -bounded. Now, for any x ∈ S, eλ .x → x (by Theorem 10.1.1(c)); since each eλ .x ∈ D.S, we have x ∈ β-cl(D.S). Hence S ⊆ β-cl(D.S) and so β-cl(S) is τ -bounded. (c) ⇒ (a). Since the module multiplication is b(A)-hypocontinuous, by Theorem 10.1.1(e), u 6 τ and so every τ -bounded set is u-bounded. Now, let S be a u-bounded set in Y . Then, if D = {eλ : λ ∈ I}, D.S is τ -bounded. [Indeed, let G ∈ W. Since D is a bounded set in A, M(D, G) is a uneighborhood of 0 in Y and so there exists r > 0 such that S ⊆ rM(D, G) = M(D, rG); that is, D.S ⊆ rG.] By (c), β-cl(D.S) is τ -bounded. But, again by Theorem 10.1.1(c), S ⊆ β-cl(D.S); hence S is τ -bounded. We now consider a characterization of the equivalence of the topologies β, u, and τ on Y . Theorem 10.1.6. [Kh07] Let (Y, τ ) be a topological left A-module with Y an F -space and A an F -algebra. Consider the following conditions: (a) There exists a ∈ A such that the map x → a.x is an isomorphism of Y onto Y . (b) β = τ on Y . (c) (Y, β) is metrizable. (d) (Y, β) is ultrabornological. (e) β = u on Y . u (f ) For each x ∈ Y, eλ .x → x. Then (a) ⇒ (b) ⇒ (c) ⇒ (d) ⇒ (e); if A has a two-sided approximate identity, (e) ⇒ (f ). If (Y, τ ) is a Banach algebra and A has a bounded two-sided approximate identity, then (f ) ⇒ (a); that is, (a) − (f ) are equivalent: β

Proof. (a) ⇒ (b). Suppose (a) holds, and let xα → x in Y . Now φ : x → a.x is a continuous, linear and one-one map from (Y, τ ) onto (Y, τ ). As a consequence of the open mapping theorem, the inverse map

174

10. THE GENERAL STRICT TOPOLOGY τ

τ

φ−1 : a.x → x is continuous. Since a.xα → a.x, we have φ−1 (a.xα ) → τ φ−1 (a.x); that is, xα → x. So τ 6 β. (b) ⇒ (c). This is obvious since (Y, τ ) is metrizable. (c) ⇒ (d) since every metrizable TVS is ultrabornological. (d) ⇒ (e). This follows from Theorem 10.1.4(ii). (e) ⇒ (f). This follows from Theorem 10.1.1(c). Now suppose that (Y, τ ) is a Banach algebra and A has a bounded two-sided approximate identity. Then (f) ⇒ (a) follows from ([ST69], Theorem 2.4). Remark. We can apply above results to extend an observation on inner derivation, due to Phillips [Ph95]. Let (Y, τ ) be a topological algebra and A a closed two-sided ideal in Y, with Y not necessarily an F -algebra. A linear map D : A → A is called a derivation on A if D(ab) = D(a)b + aD(b),

a, b ∈ A.

For any x ∈ Y, the map δx : A → A defined by δx (a) = [x, a] = xa − ax, a ∈ A, is a derivation on A, called the inner derivation on A. A derivation D : A → A is called approximately inner if there exists a net {aα : α ∈ J} in A such that D(a) = limα δaα (a) for all a ∈ A. We claim that, for any x ∈ Y, δx is approximately inner. Indeed, by Theorem 10.1.1 (d), A is β-dense in Y and so there exists a net {aα : α ∈ J} in A such β that aα → x. Then, for any a ∈ A, δx (a) = lim(aα a − aaα ) = lim δaα (a). α

α

This was observed earlier in ([Ph95], p. 244) in the case of Y = Md (A). We now consider the notion of essential modules, due to Rieffel [Rie67]. Definition. Let (Y, τ ) be a topological left A-module with A having a left approximate identity {eλ }. The essential part Ye of Y is defined as τ Ye = {x ∈ Y : eλ .x −→ x}. We say that Y is essential if Y = Ye . Clearly, Ye is a topological left A-submodule of Y. Note that, for any τ a ∈ A and x ∈ Y, eλ .(a.x) = (eλ a).x −→ a.x (by separate continuity of module multiplication) and so a.x ∈ Ye ; hence [A.Y ] ⊆ Ye , where [A.Y ] is the vector subspace of Y spanned by A.Y = {a.x :∈ A, x ∈ Y }. Further, τ for any x ∈ Ye , clearly {eλ .x : λ ∈ I} ⊆ A.Y and eλ .x −→ x and so x ∈ τ -cl[A.Y ]. Thus [A.Y ] ⊆ Ye ⊆ τ -cl[A.Y ].


175

We can now state an analogue of factorization theorem 10.1.1 for topological modules, again due to Ansari-Piri [AP90]. A topological left A-module Y is called A-factorable if, for each x ∈ Y, there exist a ∈ A and y ∈ Y such that x = a.y. Theorem 10.1.7. [AP90] Let A be a fundamental F -algebra with a uniformly bounded left approximate identity. If Y is an F -space which is an essential topological left A-module, then Y is A-factorable. We mention that if Y is A-factorable, then Y is essential since Y = A.Y ⊆ Ye ⊆ Y, or that Y = Ye . Note that the above is a so-called lefthand version of the factorization theorem. Clearly its right-hand version also holds. We next consider some properties of the essential part Ye of Y Theorem 10.1.8. [Kh07] Let (Y, τ ) be a topological left A-module with Y ultrabarralled and A having a bounded left approximate identity {eλ : λ ∈ I}. Then: (i) Ye is τ -closed in Y. (ii) Ye = τ -cl[A.Y ]. τ Proof. (i) Let x ∈ τ -cl(Ye ). We need to show that eλ .x −→ x. Let G ∈ W. Choose a balanced H ∈ W such that H + H + H ⊆ G. For each λ ∈ I, define Lλ : Y → Y by Lλ (y) = Leλ (y) = eλ .y, y ∈ Y . Since D = {eλ : λ ∈ I} is bounded in A, it follows from the proof of Lemma 10.1.3 that {Lλ : λ ∈ I} is pointwise bounded and hence equicontinuous in CL(Y, Y ) by the principle of uniform boundedness. There exists H1 ∈ W such that Lλ (H1 ) ⊆ H for all λ ∈ I. Since x ∈ τ cl(Ye ), we can choose xo ∈ Ye such that x − xo ∈ H1 ∩ H. Since eλ .xo → xo , choose λo ∈ I such that eλ .xo − xo ∈ H for all λ ≥ λo . Hence, for any λ ≥ λo , eλ .x − x = eλ .(x − xo ) + (eλ .xo − xo ) + (xo − x) ∈ Lλ (H1 ∩ H) + H + H1 ∩ H ⊆ H + H + H ⊆ G; τ

that is, eλ .x −→ x and so x ∈ Ye . Hence Ye is τ -closed. (ii) As seen above, [A.Y ] ⊆ Ye ⊆ τ -cl[A.Y ]. Hence, by (i), Ye = τ

176


-cl[A.Y ].

Corollary 10.1.9. Let Y and A be as above. Then Y is essential τ iff Y = [A.Y ] . τ Proof. By definition, Y is essential if Y = {x ∈ Y : eλ .x −→ x}. τ Hence, by above theorem, Y is essential iff Y = [A.Y ] . We mention that if A is a fundamental F -algebra with a uniformly bounded left approximate identity and Y is an F -space which is an essential topological left A-module, then, by Theorem 10.1.7, Y is A-factorable and so Y = A.Y ⊆ Ye ⊆ Y, or thatY = A.Y = Ye . Following Rieffel ([Rie67], p. 454), we may also consider a version of the Theorem 10.1.8 for the u-topology on Y , as follows. Theorem 10.1.10. [Kh07] Let (Y, τ ) be an essential topological left A-module with b(A)-hypocontinuous module multiplication and A having a bounded left approximate identity {eλ : λ ∈ I}. Then u-cl[A.Y ] = {x ∈ u Y : eλ .x −→ x}. u

Proof. Let Y1 = {x ∈ Y : eλ .x −→ x}. For any y ∈ Y1 , each u eλ .y ∈ A.Y1 ⊆ A.Y and eλ .y −→ y, and so clearly y ∈ u-cl[A.Y ]; hence Y1 ⊆ u-cl[A.Y ]. Now, let x ∈ u-cl[A.Y ]. Then there exists a net {xα : u u α ∈ J} in [A.Y ] such that xα −→ x. To show that eλ .x −→ x, let D be a bounded subset of A and G ∈ W. Choose balanced H ∈ W such that H + H + H ⊆ G. By b(A)-hypocontinuity, choose H1 ∈ W with D.H1 ⊆ H. Put D1 = D ∪ {eλ : λ ∈ I}, a bounded set in A. Since u xα −→ x, choose αo ∈ J such that xα − x ∈ M(D1 , H1 ∩ H) for all α ≥ αo . τ

Since (Y, τ ) is essential, eλ .xαo −→ xαo and so there exists λo ∈ I such that eλ .xαo − xαo ∈ H1 for all λ ≥ λo . Hence, for any a ∈ D and λ ≥ λo , a.(eλ .x − x) = aeλ .(x − xαo ) + a.(eλ .xαo − xαo ) + a.(xαo − x) ∈ a.(H1 ∩ H) + a.H1 + H1 ∩ H ⊆ H + H + H ⊆ G; u

that is, eλ .x −→ x, and so x ∈ Y1 . Thus u-cl[A.Y ] = Y1 .

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2. Essentiality of Modules of Vector-valued Functions We now present two results on essentiality of topological left Amodules of continuous vector-valued functions endowed with the uniform and compact-open type topologies. Recall that if X be a Hausdorff topological space and E a non-trivial Hausdorff TVS with WE a base of neighborhoods of 0 in E, then the uniform topology u on Cb (X, E) is the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form N(X, W ) = {f ∈ Cb (X, E) : f (X) ⊆ W }, where W varies over WE . Theorem 10.2.1. [Kh07] Suppose A is a topological algebra with a bounded approximate identity {eλ : λ ∈ I} and E a topological left Amodules such that the module multiplication is both b(A)-hypocontinuous and b(E)-hypocontinuous. Then: (i) (Cb (X, E), u) and (Co (X, E), u) are topological left A-modules with respect to the module multiplication (a, f ) → a.f as pointwise action: (a.f )(x) = a.f (x), a ∈ A, f ∈ Cb (X, E), x ∈ X. (ii) If X is locally compact, then (Co (X, E), u) is essential. Proof. (i) First, we need to show that if {aα : α ∈ J} ⊆ A with u aα → 0 and f (6= 0) ∈ Cb (X, E), then aα .f → 0 in Cb (X, E). Let G ∈ WE . Since f (X) is bounded in E, by b(E)-hypocontinuity, there exists a neighborhood V of 0 in A such that V.f (X) ⊆ G. Since aα → 0 in A, there exists αo ∈ J such that aα ∈ U for all α ≥ αo . Hence aα .f (X) ⊆ V.f (X) ⊆ G or aα .f ∈ N(X, G) for all α ≥ αo . u

Next, let a (6= 0) ∈ A and {fα : α ∈ J} ⊆ Cb (X, E) with fα → 0 u in Cb (X, E). Then a.fα → 0 in Cb (X, E), as follows. Let G ∈ WE . By b(A)-hypocontinuity, there exists an H ∈ WE such that aH ⊆ G. Since u fα → 0 in Cb (X, E), there exists an αo ∈ J such that fα (X) ⊆ H for all α ≥ αo . Hence a.fα ∈ N(X, aH) ⊆ N(X, G) for all α ≥ αo . u

(ii) Fix f ∈ Co (X, E). We need to show that eλ .f → f. Let G ∈ WE . Choose a balanced H ∈ WE with H +H +H ⊆ G. Since D = {eλ : λ ∈ I} is bounded in A, by b(A)-hypocontinuity, there exists V ∈ WE such that D.V ⊆ H.

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Since f ∈ Co (X, E), there exists a compact set K ⊆ X such that f (x) ∈ V ∩ H for all x ∈ X\K. Since f (K) is compact, there exists a finite set {z1 , ..., zn } ⊆ K such that f (K) ⊆ ∪ni=1 (f (zi ) + V ∩ H). Since E is an essential A-module, there exists λo ∈ I such that eλ f (zi ) − f (zi ) ∈ H for all λ ≥ λo and i = 1, ..., n. Now, let x ∈ X and λ ≥ λo . First suppose x ∈ K. Choose i ∈ {1, ..., n} such that f (x) − f (zi ) ∈ V ∩ H. Then eλ f (x) − f (x) = eλ (f (x) − f (zi )) + (eλ f (zi ) − f (zi )) + (f (zi ) − f (x)) ∈ D.(V ∩ H) + H − (V ∩ H) ⊆ H + H + H ⊆ G. If x ∈ X\K, eλ f (x) − f (x) ∈ eλ .(V ∩ H) − V ∩ H ⊆ eλ .V − H ⊆ H + H ⊆ G.

Definition. If X is a completely regular hemicompact space with {Kn } an increasing sequence of compact sets which form a base for compact sets on X, we define the compact-open type topology k on C(X, E) as the linear topology which has a base of neighborhoods of 0 consisting of all sets of the form N(Kn , W ) = {f ∈ C(X, E) : f (Kn ) ⊆ W }, where n ≥ 1and W ∈ WE . Theorem 10.2.2. [Kh07] Let X, E and k be as in the above definition. Suppose A is a topological algebra with a bounded approximate identity {eλ : λ ∈ I} and E a topological left A-module such that the module multiplication is both b(A)- and b(E)- hypocontinuous. Then: (i) (C(X, E), k) is a topological left A-module with respect to the module multiplication (a, f ) → a.f. (ii) (C(X, E), k) is essential. Proof. (i) First, we need to show that if {bα : α ∈ J} ⊆ A with bα → 0 k and f (6= 0) ∈ C(X, E), then bα .f → 0 in C(X, E). Fix any Km ∈ {Kn } and G ∈ WE . By b(E)-hypocontinuity, there exists a neighborhood V of 0 in A with V.f (Km ) ⊆ G. Since bα → 0 in A, there exists αo ∈ J such that

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bα ∈ V for all α ≥ αo . Hence, for any α ≥ αo , bα .f (Km ) ⊆ V.f (Km ) ⊆ G and so bα .f (Km ) ⊆ V.f (Km ) ⊆ G and so bα .f ∈ N(Km , G). k

Next, let b (6= 0) ∈ A and {gα : α ∈ J} ⊆ C(X, E) with gα → 0 in k C(X, E). Then b.gα → 0 in C(X, E), as follows. Fix any Km ∈ {Kn } and G ∈ WE . By b(A)-hypocontinuity, there exists H ∈ WE with b.H ⊆ G. Choose αo ∈ J such that gα (Km ) ⊆ H for all α ≥ αo . Hence, for any α ≥ αo , b.gα (Km ) ⊆ b.H ⊆ G and so b.gα ∈ N(Km , b.H) ⊆ N(X, G). k

(ii) Fix f ∈ C(X, E). We need to show that eλ .f → f. Let Km ∈ {Kn } andG ∈ WE . Choose a balanced H ∈ WE with H + H + H ⊆ G. Since D = {eλ : λ ∈ I} is bounded in A, by b(A)-hypocontinuity, there exists V ∈ WE such that D.V ⊆ H. Since f (Km ) is compact, there exists a finite set {z1 , ..., zn } ⊆ Km such that f (Km ) ⊆ ∪ni=1 (f (zi ) + V ∩ H). Since E is an essential A-module, there exists λo ∈ I such that eλ f (zi ) − f (zi ) ∈ H for all λ ≥ λo and i = 1, ..., n. Now, let x ∈ Km and λ ≥ λo . Choose i ∈ {1, ..., n} such that f (x) − f (zi ) ∈ V ∩ H. Then eλ f (x) − f (x) = eλ .(f (x) − f (zi )) + (eλ .f (zi ) − f (zi )) + (f (zi ) − f (x)) ∈ D.(V ∩ H) + H + V ∩ H ⊆ H + H + H ⊆ G.

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3. Topological Modules of Homomorphisms In this section, we shall study the properties of the topological module HomA (A, Y ) of continuous homomorphisms. These results were originally given in the papers [Rie67, ST69, Shan04] and in the present general form in [Kh08]. Definition: Let E and Y be topological left A-modules, where E and Y are TVSs and A a topological algebra. Then a mapping T : E → Y is called an A-module homomorphism if T (a.x) = a.T (x) for all a ∈ A and x ∈ E ([Rie67], p. 447). (Similarly, if E and Y are right A-modules, then we can define an A-module homomorphism as a mapping T : E → Y satisfying T (x.a) = T (x).a for all a ∈ A and x ∈ E. We will state results for left modules over A, similar results holding, of course, for right modules.) Our main interest is the study of A-module homomorphism from A into Y . Lemma 10.3.1. [Kh08a] Let Y be a left A-module. Suppose that A is faithful in Y . Then any A-module homomorphism T : A → Y is homogeneous (that is, T (λa) = λT (a) for all λ ∈ K and a ∈ A). Proof. Let a ∈ A and λ ∈ K. Then, for any c ∈ A, c.T (λa) = T (c.(λa)) = T ((λc)a) = (λc).T (a) = c.λT (a) Since A is faithful in Y, T (λa) = λT (a). We now establish the linearity and continuity of an A-module homomorphisms using the factorization theorem. The proof of the following theorem is analogous to those given in [Jo66, Cra69, Rie69, SumM72, KMT99]. Theorem 10.3.2. [Kh08a] Let Y be a topological left A-module with Y metrizable and A strongly factorable. Then any A-module homomorphism T : A → Y is linear and continuous. Proof. To show that T is linear, let a1 , a2 ∈ A and α, β ∈ K. If we take {an } = {a1 , a2 , 0, 0, ...}, then clearly an → 0; since A is strongly factorable, there exist b, c1 , c2 ∈ A such that a1 = c1 b, a2 = c2 b. So T (αa1 + βa2 ) = T ((αc1 + βc2 )b) = (αc1 + βc2 ).T (b) = αT (c1b) + βT (c2 b) = αT (a1 ) + β(a2 ); hence T is linear. Since Y is metrizable, to show that T is continuous, it suffices to show that if {an } ⊆ A with an → 0, then T (an ) → 0. Using

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again the strong factorability of A, we can write an = cn b, where b ∈ A and {cn } ⊆ A with cn → 0. Then T (an ) = T (cn b) = cn .T (b) → 0.T (b) = 0. (by the separate continuity of module multiplication). Thus T is continuous. Definition. (Cf. [Rie67], p. 447) Let E and Y be topological left A-modules, where E and Y are TVSs and A a topological algebra. Let HomA (E, Y ) denote the vector space of all continuous linear left Amodule homomorphisms of E into Y. If E is an A-bimodule, then defining (a ∗ T )(x) = T (x.a), HomA (E, Y ) becomes a left A-module. In fact, for any b ∈ A, x ∈ E, (a ∗ T )(b.x) = T ((b.x).a) = T (b.(x.a)) = b.T (x.a) = b.(a ∗ T )(x). In particular, HomA (A, Y ) is a left A-module. Note that if A is commutative, then defining (T ∗ a)(x) = T (a.x), HomA (E, Y ) becomes a right A-module. In fact, for any b ∈ A, x ∈ E, (T ∗ a)(b.x) = T (a.(b.x)) = T (b.(a.x)) = b.T (a.x) = b.(T ∗ a)(x). Lemma 10.3.3. [Kh08a] Let E and Y be topological left A-modules with A having an approximate identity {eλ : λ ∈ I}. If E is an essential A-module, then HomA (E, Y ) = HomA (E, Ye ). In particular, HomA (A, Y ) = HomA (A, Ye ). Proof. Since Ye ⊆ Y, clearly HomA (E, Ye ) ⊆ HomA (E, Y ). Now let T ∈ HomA (E, Y ). Then, for any x ∈ E, since eλ .x → x, lim eλ .T (x) = lim T (eλ .x) = T (x). λ

λ

Therefore T (x) ∈ Ye , i.e. T ∈ HomA (E, Ye ). We mention that HomA (E, Y ) has been extensively studied in the case of E and Y as the Banach modules of Banach-valued function spaces L1 (G, A) and Co (G, A), where G is a locally compact abelian group and A a commutative Banach algebra (see, e.g., [Wen52, Rie69, HL92]. If E = Y = A, then HomA (A, A) is the usual multiplier algebra of A, and is denoted by M(A) (see section 10.1). Definition: Let A be a Hausdorff topological algebra and (Y, τ ) a Hausdorff TVS which is a topological left A-module and has a base WY of neighborhoods of 0 in Y . The topology of bounded convergence tu = tuA (resp. the topology of pointwise convergence tp = tpA ) on HomA (A, Y )

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is defined as the linear topology which has a base of neighborhood of 0 consisting of all sets of the form U(D, G) = {T ∈ HomA (A, Y ) : T (D) ⊆ G}, where D is a bounded (respectively, finite subset) of A and G ∈ WY . Clearly, tp 6 tu . Further, we obtain: Lemma 10.3.4. [Kh08a] Let (Y, τ ) be a topological left A-module with b(A)-hypocontinuous module multiplication. Then both (HomA (A, Y ), tu ) and (HomA (A, Y ), tp ) are topological left A-modules. Proof. We prove the result only for (HomA (A, Y ), tu ). For any a ∈ A and T ∈ HomA (A, Y ), the map (a, T ) → a ∗ T is separately continuous, as follows. First, let {aα : α ∈ J} be a net in A with aα → a ∈ A, and let D be a bounded subset of A and G ∈ WY . By b(A)-hypocontinuity, there exists a balanced H ∈ WY such that D.H ⊆ G. Since T is continuous, there exists αo ∈ J such that T (aα ) − T (a) ∈ H for all α ≥ αo . Now, for any b ∈ D and α ≥ αo , (aα ∗ T − a ∗ T )(b) = T (baα ) − T (ba) = b.[T (aα ) − T (a)] ∈ D.H ⊆ G; t

u that is, aα ∗ T − a ∗ T ∈ U(D, G) for all α ≥ αo . Hence aα ∗ T → a ∗ T. tu Now, let {Tα : α ∈ J} be a net in HomA (A, Y ) with Tα → T ∈ HomA (A, Y ), and let D be a bounded subset of A and G ∈ WY . Since the map Ra : A → A given by Ra (b) = ba, b ∈ A, is linear and continuous (by separate continuity of multiplication in A), it follows that Ra (D) = Da tu is bounded in A. Since Tα → T , there exists αo ∈ J such that

Tα − T ∈ U(Da, G) for all α ≥ αo . Now, for any b ∈ D and α ≥ αo , (a ∗ Tα − a ∗ T )(b) = (Tα − T )(ba) ∈ G; t

u that is, a ∗ Tα − a ∗ T ∈ U(D, G) for all α ≥ αo . Hence a ∗ Tα → a ∗ T. The following results generalize some results of [Wan61, Bus6, Pom92, KMT99] to modules of continuous homomorphisms.

Theorem 10.3.5. [Kh08a] Let Y be a topological left A-module. Then: (i) If Y is an F-space and A is strongly factorable, then both (HomA (A, Y ), tu ) and (HomA (A, Y ), tp ) are complete.


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(ii) If Y is complete and A is ultrabarrelled having a bounded approximate identity, then both (HomA (A, Y ), tp ) and (HomA (A, Ye ), tp ) are complete. Proof. (i) Let {Tα : α ∈ J} be a tu -Cauchy net in HomA (A, Y ). Since tp 6 tu , {Tα : α ∈ J} is an tp -Cauchy net in HomA (A, Y ); in particular, for each a ∈ A, {Tα (a)} is a Cauchy net in X. Consequently, by completeness of Y , the mappings T : A → Y , given by T (a) = limα Tα (a) (a ∈ A), is well-defined. Further, for any a, b ∈ A, T (ab) = lim Tα (ab) = lim a.Tα (b) = a.T (b). α

α

Since Y is metrizable and A is strongly factorable, by Theorem 10.3.2, tu T ∈ HomA (A, Y ). We now show that Tα −→ T . Let D be a bounded subset of A and take closed G ∈ WY . There exists an index αo such that Tα (a) − Tγ (a) ∈ G for all a ∈ D and α, γ ≥ αo . Since G is closed, fixing α ≥ αo and taking lim, we have γ

Tα (a) − T (a) ∈ G for all a ∈ D. Hence, Tα − T ∈ U(D, G) for all α ≥ αo . Thus (HomA (A, Y ), tu ) is complete. By a similar argument, (HomA (A, Y ), tp ) is also complete. (ii) We first show that HomA (A, Y ) is tp -complete. [Let {Tα : α ∈ J} be a tp -Cauchy net in HomA (A, Y ). Then, for each a ∈ A, {Tα (a) : α ∈ J} is a Cauchy net in Ye and hence in Y for all a ∈ A. Since (Y, τ ) is complete, {Tα (a)} is convergent for all a ∈ A. Define T : A → Y by T (a) = lim Tα (a)}, a ∈ A. α

Then T is a left A-module homomorphism. Also {Tα : α ∈ J} is pointwise bounded. Since A is ultrabarrelled, by the principle of uniform boundedness, {Tα : α ∈ J} is equicontinuous. Therefore, given any closed G ∈ WY , there exists a neighborhood V of 0 in A such that Tα (V ) ⊆ G for all α ∈ J; that is ∪α∈J Tα (V ) ⊆ G. If a ∈ V, Tα (a) ∈ Tα (V ) ⊆ G. Therefore, {Tα (a) : α ∈ J} is a net in G for all a ∈ V. Thus T (a) = lim Tα (a) ∈ cl-G = G; that is, T (V ) ⊆ G. α

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Therefore T : A → Y is continuous, and so T ∈ HomA (A, Y ). Thus HomA (A, Y ) is tp -complete.] Since A has an approximate identity, by Lemma 10.3.3, HomA (A, Y ) = HomA (A, Ye ). Hence HomA (A, Ye ) is also tp -complete. Definition: For any x ∈ Y, consider the map Rx : A → Y given by Rx (a) = a.x, a ∈ A. Clearly, Rx is linear and continuous (by separate continuity of module multiplication); further, for any a, b ∈ A, Rx (ab) = (ab).x = a.(b.x) = a.Rx (b), so that Rx ∈ HomA (A, Y ). Now define a map µ : Y → HomA (A, Y ) by x ∈ Y.

µ(x) = Rx ,

It is easily seen that µ(Y ) = {Rx : x ∈ Y } is a left A-submodule of HomA (A, Y ). [In fact, for any a ∈ A and x ∈ Y, (a ∗ Rx )(b) = Rx (ba) = b.(a.x) = Ra.x (b), b ∈ A, so that a ∗ Rx ∈ HomA (A, Y ).] Theorem 10.3.6. [Kh08a] Let Y be a topological left A-module with A having a two-sided approximate identity {eλ : λ ∈ I}. Then µ(Ye ) is tp -dense in HomA (A, Ye ); in particular, µ(Y ) is tp -dense in HomA (A, Ye ). Proof. Let T ∈ HomA (A, Ye ). For each λ ∈ I, define xλ = T (eλ ). Then lim eγ .xλ = lim eγ .T (eλ ) = lim T (eγ eλ ) = T (eλ ) = xλ , γ

γ

γ

and so xλ ∈ Ye . Now, for any a ∈ A, a.xλ = a.T (eλ ) = T (aeλ ) → T (a); hence µ(xλ )(a) = Rxλ (a) = a.xλ = a.T (eλ ) = T (a.eλ ) → T (a) tp

Therefore µ(xλ ) → T. Thus T ∈ tp -cl[µ(Ye )]; that is, µ(Ye ) is tp -dense in HomA (A, Ye ). Remark: We mention that µ(Y ) need not be tu -closed in HomA (A, Y ) even if Y = A is a metrizable locally C ∗ -algebra or a Banach algebra (see also ([Lar70], p. 17) and ([Ph88], p.187)). We include the following example, due to Wang ([Wan61], p. 1138), shows that µd (A) need not be u-closed in CMd (A), in general. (See also ([Lar70], p. 17) and ([Ph88], p.187)).


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Example. Let ϕ : R → R be a function with ϕ(x) ≥ 1 (x ∈ R) and lim ϕ(x) = lim ϕ(x) = +∞.

x→+∞

x→−∞

Define A = Aϕ = {f ∈ C(R, C) : kf kϕ = sup | ϕ(x)f (x) |< ∞}. x∈R

Then A forms a commutative normed algebra under pointwise operations and the above norm. To see that A is complete, let {fn } be a Cauchy sequence in A. Then {fn } and {fn ϕ} are Cauchy sequences in the usual supremum norm k.k∞ . Let f and g be, respectively, the uniform limits of {fn } and {fn ϕ}. Then sup | g(x) |< ∞ and f ϕ = g. x∈R

Hence A is a Banach algebra. Further, A is faithful. It is apparent that A ⊆ Co (R, C) but A 6= p Co (R, C). Indeed any real-valued function f ∈ Co (R, C) such that f (x) ϕ(x) ≥ 1 belongs to Co (R, C)\ A. However, A with the operator norm is dense in Co (R, C), and so A is not complete in the operator norm. For instance, let f ∈ Co (R, C) and, since Coo (R, C) is dense in Co (R, C), there exists a sequence {fn } ⊆ A of functions with compact support such that lim kfn − f k∞ = 0.

n→∞

Then sup kfn g − f gkϕ = kgk≤1

sup sup | (fn (x) − f (x))g(x)ϕ(x) | kgk≤1 x∈R

≤ kfn − f k∞ sup sup | g(x)ϕ(x) | kgk≤1 x∈R

≤ kfn − f k∞ , showing that f ∈ cl-(A). Consequently, A is not closed in Co (R, C) in the operator norm. Theorem 10.3.7. Let X be a topological left A-module with A having a two-sided approximate identity {eλ : λ ∈ I}. If (X, β) is complete, the map µ : X → HomA (A, Xe ) defined by µ(y) = Ry , y ∈ X, is onto. Proof. By Theorem 10.3.6, µ(X) is tp -dense in HomA (A, Xe ). We now show that µ(X) is tp -closed in HomA (A, Xe ). Let T ∈ tp -cl µ(X). tp

There exists a net {xα : α ∈ J} ⊆ X such that Rxα → T. Then {xα : α ∈ J} is β-Cauchy in X. [Let D be a finite subset of A and G ∈ WX .

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10. THE GENERAL STRICT TOPOLOGY tp

Choose a balanced H ∈ WX with H + H ⊆ G. Since Rxα → T, there exists αo ∈ I such that for all α ≥ αo , Rxα − T ∈ N(D, H) or Rxα (a) − T (a) ∈ H for all a ∈ D. Then, for any a ∈ D and α, γ ≥ αo , a.xα − a.xγ = [Rxα (a) − T (a)] + [T (a) − Rxγ (a)] ∈ H + H ⊆ G.] β

tp

Since (X, β) is complete, xα → xo , xo ∈ X. Hence Rxα → Rxo . By uniqueness of limit in Hausdorff spaces, T = Rxo ∈ µ(X). Thus µ(X) = HomA (A, Xe ).


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4. Notes and Comments Section 10.1. In 1968, Busby [Bus68] also defined a general strict topology on any B ∗ -algebra Y which contains A as a closed two-sided ideal; this is given by the seminorms x → max{||ax||, ||ax||} (a ∈ A). Independently, this topology was studied in greater detail and in a more general setting by Sentilles and Taylor [ST69] where A is a Banach algebra having an approximate identity and Y is a left Banach A-module. More recently, Khan-Mohammad-Thaheem [KMT05] have considered the case of Y a locally idempotent F -algebra which contains A as a closed two-sided ideal, while Shantha [Sha04] has taken Y a locally convex module over a locally convex algebra A. As pointed out in Sentilles and Taylor [ST69], the essential ingredient of arguments used by Buck [Buc58] and later authors is the presence of an approximate identity in Co (X) or A. This section contains some results from the paper [Kh07] whish deal with the study of general strict and related topologies on a topological left A-module Y, where Y is Hausdorff TVS and A a Hausdorff topological algebra having a two-sided bounded approximate identity. These include generalization of several results of Sentilles and Taylor [ST69] from the setting of Banach left modules to topological left modules obtained by Shantha [Shan04, KMT05]. Section 10.2. This section two results from [Kh07] on the essentiality of topological left A-modules of continuous vector-valued functions, which were originally proved in [SumW72] for a particular case. Section 10.3. In this section, a study is made of the properties of the topological module HomA (A, Y ) of continuous homomorphisms. In [Rie67, Rie69], Rieffel made an extensive study of the Banach module HomA (A, X) of continuous homomorphisms. Further results in this direction have been obtained in [ST69, Rue77] in connection with the study of general strict topology. More recently, Shantha [Sha04, Shan05] has studied homomorphisms in the case of locally convex modules. This section contains some results from a recent work [Kh08a] which has investigated the extent to which some of the results of above authors are also true in the non-locally convex setting of topological modules. As application, an observation on inner derivation, due to Phillips [Ph95], is presented in a more general setting. We mention that particular cases of the module HomA (A, X) are the algebras Mℓ (A), Mr (A), M(A) and Md (A) which consists of all continuous left multipliers, right multipliers, multipliers and double multipliers. There is an extensive literature on multipliers; see, e.g., [Wan61,

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Jo64, Jo66, Tay70, TW70, APT73, Fon75, Lat76, Tom76, Dav77, CL78, Won85, Ph88, Hus89, KMT99, MKT01].

CHAPTER 11

Mean Value Theorem and Almost Periodicity

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11. MEAN VALUE THEOREM AND ALMOST PERIODICITY

1. The Mean Value Theorem in TVSs We present here the mean value theorem and the mean value inequality for class of Gateaux differentiable functions f : A ⊆ E → F, where E and F are TVSs (both over the field R of real numbers) and A ⊆ E an open set. These results are given in [Vain64] for E and F normed spaces and in the present general form in [Kh97]. Definition. A function f : A → F is said to be Gateaux differentiable at xo ∈ A if there exists a mapping from E into F , denoted by f ′ (xo ), such that, given any z ∈ E and a balanced neighborhood V of 0 in F , there exists a δ > 0 satisfying, f (xo + tz) − f (x) − f ′ (xo )(z) ∈ V, t whenever 0 < |t| < δ; f ′ (xo ) is called the Gateaux derivative of f at xo and we briefly write as, f (xo + tz) − f (xo ) , z ∈ E. (1) t→0 t Some authors also require f ′ (xo ) to be linear, but it is not assumed here. The following example shows that a Gateaux differentiable function needs not being continuous. f ′ (xo )(z) = lim

Example. The function f : R2 → R, given by a2 b if x = (a, b) 6= (0, 0), and f ((0, 0)) = 0, a4 + b2 is not continuous at x = (0, 0), although f ′ (0) exists. We first consider the Lagrange form of the mean value theorem for real-valued functions on E. For any x, y ∈ E, the line segment [x, y] joining x and y is defined as the set f (x) =

[x, y] = {(1 − t)x + ty : 0 ≤ t ≤ 1}. Theorem 11.1.1. [Kh97] Let E be a Hausdorff TVS over R, and let A ⊆ E be an open set. Let g : A → R be a function continuous and Gateaux differentiable at each point of a segment [xo , xo + h] in A. Then there exists a θ ∈ (0, 1) such that g(xo + h) − g(xo ) = g ′ (xo + θh)(h). Proof. Define φ : [0, 1] → R by φ(t) = g(xo + th), t ∈ [0, 1].

1. THE MEAN VALUE THEOREM IN TVSS

191

Then φ is differentiable on [0, 1]. In fact, using (1), we obtain [φ′ (t) = =

lim [g(xo + (t + ∆t)h) − g(xo + th)]/∆t

∆t→0 g ′ (xo

+ th)(h).

By the classical mean value theorem, there exists a θ ∈ (0, 1) such that φ(1) − φ(0) = φ′ (θ). Hence g(xo + h) − g(xo ) = φ(1) − φ(0) = g ′ (xo + θh)(h). An exact analogue of Theorem 11.1.1 for vector-valued (or even for complex-valued) functions need not hold as is shown below: Example. Consider the function f : [0, 2π] → C defined by f (x) = cos x + i sin x = eix , x ∈ [0, 2π]. Clearly f ′ (x) = ieix . Now f (2π) − f (0) = 1 − 1 = 0, but (2π − 0)f ′ (c) = 2πieic 6= 0 for all 0 < c < 2π. Hence f (2π) − f (0) 6= (2π − 0)f ′ (c) for all 0 < c < 2π. However, f (x) does satisfy the Mean Value Inequality for the values xo = 0, xo + h = 2π since |f (2π) − f (0)| = 0 ≤ sup{|2π − 0||ieic | : 0 ≤ c ≤ 2π} = 2π. However, for the class of these functions, the following version of the mean value theorem holds. Theorem 11.1.2. [Kh97] Let E and F be a Hausdorff topological vector spaces over R, and let A ⊆ E be an open set. Suppose F has ∗ a non-trivial (real) continuous dual F , and let f : A → F be a function continuous and Gateaux differentiable at each point of the segment ∗ [xo , xo + h] in A. Then, given u ∈ F , there exists a θ ∈ (0, 1) such that < f (xo + h) − f (xo ), u >=< f ′ (xo + θh)(h), u > . Proof. Define g : [xo , xo + h] → R by g(x) =< f (x), u >, x ∈ [xo , xo + h]. Then it follows from the linearity and continuity of u that g ′ (x)(h) =< f ′ (xo )(h), u > .

(2)

192


By Theorem 11.1.1, there exists a θ ∈ (0, 1) such that g(xo + h) − g(xo ) = g ′ (xo + θh)(h). This establishes (2). The above result clearly reduces to Theorem 11.1.1 when F = R. Theorem 11.1.3. (Mean Value Inequality) [Kh97] Under the hypothesis of Theorem 11.1.2, given a continuous seminorm p on F , there exists a θ ∈ (0, 1) such that, p(f (xo + h) − f (xo )) ≤ p(f ′ (xo + θh)).

(3)

Proof. By the analytic form of the Hahn-Banach theorem ([Ko69], ∗ Theorem 8, §.17.3), we can choose a u(6= 0) ∈ F such that < y, u >≤ p(y) for all y ∈ F and < f (xo + h) − f (xo ), u >= p(f (xo + h) − f (xo )). By Theorem 11.1.2, there exists a θ ∈ (0, 1) such that < f (xo + h) − f (xo ), u >=< f ′ (xo + θh)(h), u > . Consequently, we obtain (3). Remark. We have seen above that the function f : [0, 2π] → C defined by f (x) = cos x + i sin x = eix , x ∈ [0, 2π]. does not satisfy Theorem 11.1.1, the Mean Value Theorem, for the values xo = 0, xo + h = 2π. However, f (x) does satisfy Theorem 3, the Mean Value Inequality, for the values xo = 0, xo + h = 2π since |f (2π) − f (0)| = 0 ≤ sup{|2π − 0||ieic | : 0 ≤ c ≤ 2π} = 2π. As an application of the above theorem, we obtain: Corollary 11.1.4. [Kh97] Suppose F is locally convex and A ⊆ E an open connected set, and let f : A → F be continuous and Gateaux differentiable at each point of A. If f ′ (xo ) = 0 for each x ∈ A, then f is constant on A. Proof. Fix x ∈ A, and let B = {x ∈ A : f (x) = f (xo )}. Clearly, B is non-empty and closed in A. B is also open in A, as follows. Let x ∈ B. Choose a balanced neighborhood V of 0 in E such that U = x + V ⊆ A.

1. THE MEAN VALUE THEOREM IN TVSS

193

Then, for each y ∈ U, [x, y] ∈ U. Hence, for each y ∈ U and any continuous seminorm p on F , it follows from Theorem 11.1.3 and the hypothesis that p(f (y) − f (x)) ≤ sup (f ′ (z)(y − x)) = p(0) = 0. z∈[x,y]

Since F is locally convex and Hausdorff, f (y) = f (x) = f (xo ) for all y ∈ U. Hence U ⊆ B. Since A is connected, B = A. Thus f (x) = f (xo ) for all x ∈ A. Theorem 11.1.5. (Mean Value Inclusion) [AS67, Kh97] Suppose F is a locally convex space. Then, under the hypothesis of Theorem 2, f (xo + h) − f (x) ∈ co.(B), where B = {f ′ (xo )(h) : x ∈ [xo , xo + h]} and co.(B) is the closed convex hull of B. Proof. Suppose f (xo + h) − f (xo ) ∈ / co.(B). By ([Ko69], Theorem 2, ∗ §. 20.7), there exists a u(6= 0) ∈ F and an r ∈ R such that, < y, u >≤ r < f (xo + h) − f (xo ), u >

(4)

for all y ∈ co.(B). Define φ : [0, 1] → R by φ(t) =< f (xo + th), u >. Then φ′ (t) =< f ′ (xo + th)(h), u >. There exists a θ ∈ (0, 1) such that φ(1) − φ(0) = φ′ (θ). Then < f (xo + h) − f (xo ), u >= φ(1) − φ(0) =< f ′ (xo + θh)(h), u > which contradicts (4). Remark. Theorem 11.1.5 need not hold if F is not locally convex, or if co.(B) is replaced by B or co.(B). This follows from ([AS67], Examples 1.23-1.25).

194


2. Almost Periodic Functions with Values in a TVS In this section, we consider the concept of almost periodicity of functions having values in a topological vector space E. The results are taken mainly from [NGu84, KA11]. Definition: A subset P of R is called relatively dense in R if there exists a number ℓ > 0 such that every interval of length ℓ in R contains at least one point of P . Definition: Let (E, τ ) be TVS with a base W = WE of balanced neighborhoods of 0. A function f : R → E is called almost periodic if it is continuous and, for each W ∈ W, there exists a number ℓW > 0 such that each interval of length ℓW in R contains a point τW such that f (t + τW ) − f (t) ∈ W for all t ∈ R.

(∗)

A number τW ∈ R for which (∗) holds is called W -translation number of f. The above property says that, for each W ∈ W, the function f has a set of W -translation numbers PW,f which is relatively dense in R. The set of all almost periodic functions f : R → E is denoted by AP (R, E). For any f : R → E and a fixed h ∈ R, the h-translate of f is defined as the function fh : R → E defined by fh (t) = f (t + h), t ∈ R. We shall denote H(f ) = {fh : h ∈ R}, the set of all translates of f . Definition: Let X be a Hausdorff space and (E, d) a metrizable TVS. The uniform topology u on Cb (X, E) is defined as the topology given by the metric ρ ρ(f, g) = sup d(f (x), g(x)), f, g ∈ Cb (X, E). x∈X

Theorem 11.2.1. [NGu84, KA11] Let X be a Hausdorff space and (E, d) a metric linear space. (a) If a sequence {fn } ⊆ Cb (X, E) is u-convergent to a function f : X → E, then f ∈ Cb (X, E). (b) If {fn } is a u-Cauchy sequence in Cb (X, E) and if there is a function f : X → E such that fn (x) → f (x) for each x ∈ X, then u fn −→ f and f ∈ Cb (X, E). (c) If E is complete, then so is (Cb (X, E), u). Proof. (a) f is continuous as follows. Let xo ∈ X and W ∈ W. We show that there exists a neighborhood G of xo in X such that f (y) ∈ f (xo ) + W for all y ∈ G.

2. ALMOST PERIODIC FUNCTIONS WITH VALUES IN A TVS

195 u

Choose a balanced V ∈ W such that V + V + V ⊆ W . Since fn −→ f, there exists an integer N such that fn (x) − f (x) ∈ V for all n ≥ N and all x ∈ X.

(A)

Since fN is continuous at xo , there exists a neighborhood G of xo in X such that fN (y) ∈ fN (xo ) for all y ∈ G. (B) Then, for any y ∈ G, using (A) and (B), f (y) − f (xo ) = (f (y) − fN (y)) + (fN (y) − fN (xo )) + (fN (xo ) − f (xo )) ∈ V + V + V ⊆ W. Thus f is continuous on X. Next, f is bounded as follows. Suppose f is unbounded. Then there exists a W ∈ W such that f (X) is not contained in nW for all n ≥ 1. Hence, for each n ≥ 1, there exists xn ∈ X such that f (xn ) ∈ / nW . u Choose a balanced V ∈ W such that V + V ⊆ W . Since fn −→ f , there exists an integer K such that fn (x) − f (x) ∈ V for all n ≥ K and x ∈ X. In particular, fK (xn ) − f (xn ) ∈ V for all n ≥ 1. Since fK is bounded, there exists an integer m > 1 such that fK (X) ⊆ mV . Then f (xm ) = [f (xm ) − fK (xm )] + fK (xm ) ∈ V + mV ⊆ mV + mV ⊆ mW, a contradiction. Hence f is bounded. (b) Let {fn } be a u-Cauchy sequence in Cb (X, E). By hypothesis, for u each x ∈ X, f (x) := lim fn (x) exists in E. Then fn −→ f , as follows. n→∞

Let W ∈ W be closed. Since {fn } is u-Cauchy, there exists an integer J such that fn (x) − fm (x) ∈ W for all n, m ≥ J and x ∈ X. Fix any n ≥ J. Since W is closed and, for each x ∈ X, fm (x) −→ f (x), we have fn (x) − f (x) ∈ W for x ∈ X. Thus fn (x) − f (x) ∈ W for all n ≥ J and all x ∈ X; i.e. fn − f ∈ N(X, W ) for all n ≥ J. u

This proves that fn −→ f . By (a), f ∈ Cb (X, Y ). (c) Let {fn } be a u-Cauchy sequence in Cb (X, E). Since p ≤ u, {fn } is a p-Cauchy sequence in C(X, E); that is, for each x ∈ X, {fn (x)} is

196


a Cauchy sequence in E. Since E is complete, for each x ∈ X, f (x) := u lim fn (x) exists in E. Then, by (b), fn −→ f and that f is continuous n→∞

and bounded. Hence (Cb (X, E), u) is complete. Theorem 11.2.2. [NGu84, KA11] Let E be a TVS. Let f ∈ AP (R, E). Then: (a) f has precompact range f (R); hence f is bounded. (b) f is uniformly continuous on R. Proof. (a) Let W ∈ W. Choose a balanced V ∈ W such that V +V ⊆ W. Since f is almost periodic, there exists a number ℓ = ℓV > 0 such that each interval of length ℓ in R contains a point τV such that f (t + τV ) − f (t) ∈ V for all t ∈ R.

(1)

By continuity of f , the set f [0, ℓ] is compact in E and hence precompact. So there exist x 1 , ..., x n ∈ f [0, ℓ] such that f [0, ℓ] ∈

n [

(xi + V ).

(2)

i=1

We claim that f (R) ⊂

n S

(xi + W ). [Take an arbitrary t ∈ R. By (1),

i=1

there exists τ ∈ [−t, −t + ℓ] such that f (t + τ ) − f (t) ∈ V. By (2), choose xk ∈ {x1 , ..., xn } such that f (t + τ ) − xk ∈ V. Then f (t) − xk = [f (t) − f (t + τ )] + [f (t + τ ) − xk ] ∈ −V + V ⊆ W, and therefore f (t) ∈ x k + W .] Thus f (R) is precompact in E. (b) Let W ∈ W. Choose a balanced V ∈ W such that V +V +V ⊆ W. There exists a number ℓ = ℓV > 0 such that each interval of length ℓ in R contains a point τV such that f (t + τV ) − f (t) ∈ V for all t ∈ R.

(3)

Now f , being almost periodic, is continuous on R. Then f is uniformly continuous on the compact set [−1, 1 + ℓV ], so there exists δ = δV > 0 (we may assume 0 < δ < 1 without loss of generality) such that f (s) − f (t) ∈ V for all s, t ∈ [−1, 1 + ℓV ] with |s − t| < δ.

(4)

Let a, b ∈ R with |a − b| < δ and assume a < b. We claim that f (a) − f (b) ∈ W. [We may assume that a < b. Choose a τV ∈ [−a, −a + ℓV ] satisfying (3). Then a + τV ∈ a + [−a, −a + ℓV ] = [0, 0 + ℓV ] ⊆ [−1, 1 + ℓV ];


197

since 0 < b − a < δ < 1, b + τV ∈ b + [−a, −a + ℓV ] = [b − a, b − a + ℓV ] = [0, 1 + ℓV ] ⊆ [−1, 1 + ℓV ]. Also, |(a + τV ) − (b + τV )| = |a − b| < δ, so by (4), f (a + τV ) − f (b + τV ) ∈ V.

(5)

Therefore, by (3) and (5), f (a) − f (b) = [f (a) − f (a + τV )] + [f (a + τV ) − f (b + τV )] +[f (b + τV ) − f (b)] ∈ −V + V + V ⊆ W.] Thus f is uniformly continuous on R.

Remark. Clearly, by Theorem 11.2.2(a), AP (R, E) ⊆ Cb (R, E). Theorem 11.2.3. [NGu84, KA11] Let E be a TVS. If {fn } is a u sequence in AP (R, E) such that fn −→ f , then f ∈ AP (R, E). Proof. Clearly, by Theorem 11.2.2(b), f is continuous on R. Let W ∈ W. Choose a balanced V ∈ W such that V + V + V ⊆ W. Since u fn −→ f , there exists an integer N ≥ 1 such that such that fn (t) − f (t) ∈ V for all t ∈ R and n ≥ N.

(6)

Since fN is almost periodic, there exists a number ℓ = ℓV > 0 such that each interval of length ℓ in R contains a point τ = τV such that fN (t + τ ) − fN (t) ∈ V for all t ∈ R.

(7)

Then, by (6) and (7), for any t ∈ R f (t + τ ) − f (t) = [f (t + τ ) − fN (t + τ )] + [fN (t + τ ) − fN (t)] +[fN (t) − f (t)] ∈ −V + V + V ⊆ W. Since the set PV,fN of almost periods of fN is relatively dense, we take PV,f = PV,fN . Hence f is also almost periodic Theorem 11.2.4. [NGu84, KA11] Let E be a TVS. If f : R → E is almost periodic, then the functions (i) λf (λ ∈ K), (ii) f (t) ≡ f (−t) and (iii) fh (t) = f (t + h) (h ∈ R) are also almost periodic. Proof. (i) This is trivial if λ = 0. Suppose λ 6= 0. Let W ∈ W be balanced. Then V = λ−1 W ∈ W. Since f is almost periodic, there exists a number ℓ = ℓV > 0 such that each interval of length ℓ in R contains a point τV such that f (t + τV ) − f (t) ∈ V for all t ∈ R.

198


Then (λf )(t + τV ) − (λf )(t) = λ[f (t + τV ) − f (t)] ∈ λV = W for all t ∈ R. Hence λf is almost periodic. (ii) Let W ∈ W be balanced. There exists a number ℓ = ℓW > 0 such that each interval of length ℓ in R contains a point τ such that f (t + τ ) − f (t) ∈ W for all t ∈ R. Put s = −t; we get: f (s − τ ) − f (s) = f (−s + τ ) − f (−s) = f (t + τ ) − f (t) ∈ W. Therefore f (s − τ ) − f (s) ∈ W for every s ∈ R. This shows that f is almost periodic with −τ as a W -translation number. (iii) Let W ∈ W be balanced, and let h ∈ R. There exists a number ℓ = ℓW > 0 such that each interval of length ℓ in R contains a point τ such that f (t + τ ) − f (t) ∈ W for all t ∈ R. Replacing t by t + h, we have f (t + h + τ ) − f (t + h) ∈ W for all t ∈ R, or fh (t + τ ) − fh (t) ∈ W for all t ∈ R.

Theorem 11.2.5. [NGu84, KA11] Let E and F be a TVSs, and let f ∈ AP (R, E). If g : f (R)→F is any continuous function, then the composed function g ◦ f ∈ AP (R, F ). Proof. Let U ∈ WF be balanced. Since f (R) is compact, g is uniformly continuous on f (R) and so there exists a W ∈ WE such that g(x) − g(y) ∈ U for all x, y ∈ f (R) with x − y ∈ W. Since f is almost periodic, there exists a number ℓ = ℓW > 0 such that each interval of length ℓ in R contains a point τ such that f (t + τ ) − f (t) ∈ W for all t ∈ R. Consequently, g[f (t + τ )] − g[f (t)] ∈ U for all t ∈ R. Thus g ◦ f ∈ AP (R, F ). We now proceed to establish the Bochner’s criteria for almost periodicity. As a first step, we obtain:


199

Theorem 11.2.6. [NGu84, KA11] Let E be a TVS and f : R → E a continuous function. If the set of translates H(f ) = {fh : h ∈ R} is u-sequentially compact in Cb (R, E), then f is almost periodic. Proof. Suppose f is not almost periodic Then there exists a W ∈ W such that for every ℓ > 0, there exists an interval of length ℓ, [−a, −a + ℓ] (say) which contains no W -translation number of f. Consequently, for every h ∈ [−a, −a + ℓ], there exists th ∈ R such that f (th + h) − f (th ) ∈ / W. Let us consider h1 ∈ R and an interval (a1 , b1 ) with b1 − a1 > 2 |h1 | which 1 . Since contains no W -translation number of f . Now Let h2 = a1 +b 2 b1 − a1 b1 − a1 < h1 < , 2 2 h2 −h1 ∈ (a1 , b1 ) and therefore h2 −h1 cannot be a W -translation number of f . Let us consider another interval (a2 , b2 ) with b2 −a2 > 2(|h1 |+|h2 |), 2 ; then which contains no W -translation number of f . Let h3 = a2 +b 2 h3 − h1 , h3 − h2 ∈ (a2 , b2 ) and therefore h3 − h1 and h3 − h2 cannot be W -translation number of f . We proceed and get a sequence {hn } such that no hm − hn (m > n) is a W -translation number of f ; that is, there exists tmn ∈ R with −

f (tmn + hm − hn ) − f (tmn ) ∈ / W.

(8)

Put smn = tmn − hn . Then (8) becomes: f (smn + hm ) − f (smn + hn ) ∈ / W.

(9)

Since H(f ) is sequentially compact in Cb (R, E), there exists a subsequence {kn }of {hn }such that {f {t + kn )} is u-convergent on R. Then, for W ∈ W, there exists an N = NW ≥ 1 such that if m, n > N (we may take m > n), we have: f (t + km ) − f (t + kn ) ∈ W for every t ∈ R.

(10)

Taking t = smn in (10), we get a contradiction to (9). Thus f is almost periodic. As a converse of the above theorem, we obtain: Theorem 11.2.7. [NGu84, KA11] Let E be an F -space and f : R → E an almost periodic function. Then the set of translates H(f ) = {fh : h ∈ R} is u-compact in Cb (R, E). Proof. Since E is an F -space, by Theorem 11.2

200


.1, (Cb (R, E), u) is also an F-space. Therefore, it suffices to show that any sequence {fhn } in H(f ) has a u-Cauchy subsequence. Proof. Let S = {an } be a dense sequence in R. Since f (R) is precompact and hence relatively compact in the F-space E, we can extract from {fhn (a1 )} a convergent subsequence. Let {fh1,n } be the subsequence of {fhn } which converges at a1 . We apply the same argument as above to the sequence {fh1,n } to choose a subsequence {fh2,n } which converges at a2 . We continue the process and consider the diagonal sequence {fhn,n } which converges at each an in S. Call this last sequence by {fkn }. We claim that this sequence is u-Cauchy on R. Proof. Let to ∈ R, and let W ∈ W. Choose a balanced V ∈ W such that V + V + V + V + V ⊂ W . By almost periodicity of f , let ℓ = ℓ(V ) > 0 be such that each interval of length ℓ in R contains a point τ = τV such that f (t + τ ) − f (t) ∈ V for all t ∈ R.

(11)

By uniform continuity of f over R, there exists δ = δV > 0 such that f (t) − f (t′ ) ∈ V for every t, t′ ∈ R with |t − t′ | < δ.

(12)

We divide the interval [0, ℓ] into subintervals I1 , ..., Ir , each of length smaller than δ. Since S is dense in R, for each 1 ≤ j ≤ r, Ij contains a point bj of S and set So = {b1 , ..., br }. Since {fkn } is p-convergent on S, it follows that {fkn } is u-convergent on the finite set So . Then there exists an integer N = NV ≥ 1 such that f (bi + kn ) − f (bi + km ) ∈ V for every i = 1, ..., J and n, m > N. (13) Now, choose τo ∈ [−to , −to + ℓ] satisfying (11). Then to + τ ∈ [0, ℓ], so choose bj ∈ So such that |to + τo − bj | < δ. So, by (12), f (to + τo + kn ) − f (bj + kn ) ∈ V for every n ≥ 1. Therefore, if n, m > N, by applying (11), (13), (14) we get fkn (to ) − fkm (to ) = f (to + kn ) − f (to + km ) = [f (to + kn ) − f (to + kn + τo )] +[f (to + kn + τo ) − f (bj + kn )] +[f (bj + kn ) − f (bj + km )] +[f (bj + km ) − f (to + τo + km )] +[f (to + km + τo ) − f (to + km )] ∈ V +V +V +V +V ⊂W

(14)


201

Therefore the subsequence {fkn } of {fhn } is u-Cauchy on R. Consequently, {fkn } is u-convergent on R. Combining Theorems 11.2.6 and 11.2.7, we have: Corollary 11.2.8. (Bochner’s Criterion) [NGu84, KA11] Let E be an F -space. Then a continuous function f : R → E is almost periodic function iff the set H(f ) = {fh : h ∈ R} is u-compact in Cb (R, E). Theorem 11.2.9. Let E be an F -space, and let f1 , ..., fm ∈ AP (R, E). Define F : R → E m by F (t) = (f1 (t), ..., , fm (t)), t ∈ R. Then F ∈ AP (R, E m ). In particular, for any W ∈ W, f1 , ..., fm have common W -translation numbers. Proof. Let {hn } be a sequence in R. Consider the sequence {f1,hn } of translates of function f1 corresponding to {hn }. Since f1 is almost periodic, by using Bochner’s criteria, we can extract from {f1,hn } a uniformly convergent subsequence, denoting again by {f1,hn }. Continuing this process, we extract from {fm,hn } a uniformly convergent subsequence, denoted also by {fm,hn }. Then the sequence {(f1,hn ..., fm,hn )} has a subsequence which is easily seen to be u-convergent on R. Hence F is almost periodic. Next, for any W ∈ W, there exists a number ℓ = ℓW > 0 such that each interval of length ℓ in R contains a point τW such that F (t + τW ) − F (t) ∈ W m for all t ∈ R. Consequently, fi (t + τW ) − fi (t) ∈ W for all t ∈ R and i = 1, ..., m. Theorem 11.2.10. [NGu84, KA11] Let E be a F -space, and let f, g ∈ AP (R, E). Then f + g ∈ AP (R, E). Proof. In view of the Bochner’s criteria, we need to show that H(f + g) = {fh + gh : h ∈ R} is u-compact in Cb (R, E). Define F : R → E 2 by F (t) = (f (t), g(t)), t ∈ R. By Theorem 11.2.8, F ∈ AP (R, E 2 ) and hence, by Bochner’s criteria, H(F ) = {(fh , gh ) : h ∈ R} is u-compact in Cb (R, E 2 ). Now define S : Cb (R, E) × Cb (R, E) → Cb (R, E) by S(u, v) = u + v, u, v ∈ Cb (R, E). This is a continuous function, hence S(H(F )) = H(f + g)) is u-compact in Cb (R, E). We next consider completeness of the space (AP (R, E), u).

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Theorem 11.2.11. [NGu84, KA11] Let E be an F -space. Then the vector space AP (R, E) is u-complete. Proof. By Theorems 11.2.4. and 11.2.10, AP (R, E) is a vector space. Further, since each f ∈ AP (R, E) is bounded, AP (R, E) is a vector subspace of Cb (R, E). Since E is complete, by Theorem 11.2.1, Cb (R, E) is u-complete. So we need only to see that AP (R, E) is u-closed in Cb (R, E). Proof. Let f ∈ Cb (R, E) with f ∈ u-cl[AP (R, E)]. Then there exists u a sequence {fn } ⊆ AP (R, E) such that fn −→ f . Since each fn is almost u periodic and fn −→ f , by Theorem 11.2.3, f is almost periodic. Hence f ∈ AP (R, E). Thus AP (R, E) is u-closed in Cb (R, E). Remark. We mention that a continuous functions f : [a; b] → E need not be integrable in the Riemann sense, if E is not locally convex ([Rol85], p. 123). However, if E is a locally pseudoconvex F-space, then all analytic functions f : [a; b] → E are integrable in the Bochner-Lebesgue sense ([Rol85], Theorem 3.5.2). Using a similar approach, almost periodicity of functions with values in p-Fréchet spaces, 0 < p < 1, has been considered in [GaNGu07]. This paper also contains some applications to differential equations and to dynamical systems.


203

3. Notes and Comments Section 11.1. A useful reference for the background of the results in this section is [FM91]; see also [Mcl65, Vain64]. These results are given in [Vain64] for E and F normed spaces and in [AS67] for E and F locally convex space. In the general case, similar results are given in [Llo73] without proof. The proofs in the present form are taken from [Kh97]. These include a shorter and direct proof of Theorem 11.5 based on the separation form of the Hahn-Banach theorem (cf. [AS67]). The reader is referred to [PRS01] for more recent versions of the theorem in the TVS setting. We now indicate further possible generalizations of the mean value theorem. (I) For set-valued mappings. In this case, two useful references are the papers: (i) P.H. Sach: Differentiation of set-valued maps in Banach spaces, Math. Nachr. 139(1988), 215-235. (ii) P.H. Sach and N.Q. Lan, A mean value theorem for set-valued maps, Rev. Roumaine Math. Pures Appl. 36(1993), 359-368. These papers contain versions of the Mean Value Theorem in the Banach space setting and it may be possible to extend these results to the setting of topological vector spaces. (II) For Fuzzy mappings. A useful references in this direction is the paper: M. Ferraro and D.H. Foster, Differentiation of fuzzy continuous mappings on fuzzy topological vector spaces, J. Math. Anal. Appl., 121(1987), 589-601. This paper does not contain any version of the Mean Value Theorem, but it seems likely that such a theorem can be formulated. Section 11.2. The theory of almost periodic functions was mainly created and published during 1924-1926 by the Danish mathematician Harold Bohr; Bohr’s work was preceded by the important investigations of P. Bohl and E. Esclangon (see references in [Cor89, LeZh82]). The theory attracted the interest of a number of researchers because it has been effectively applied to solutions of diverse problems, initially in the theory of harmonic analysis and differential equations. In 1933, S. Bochner [Bo33] published an important article devoted to the extension of the theory of almost periodic functions on the real line R with values in a Banach space E. His results were further developed by several mathematicians (see the monographs by L. Amerio and G. Prouse [AmPr71], C. Corduneanu [Cor89], B.M. Levitan and V.V. Zhikov [LeZh82], and

204


S. Zaidman [Za85]). The theory of almost periodic functions taking values in a complete metrizable locally convex space E has been initiated by G.M. N’Guérékata [NGu84] and further developed in [BuNgu04, GaNGu07, NGu01]. A survey paper by A.I. Shtern [Sh05] also considers almost periodic functions and representations in locally convex spaces. In this section, we have considered the concept of almost periodicity of functions having values in a topological vector space E, not necessarily locally convex. We are mainly concerned with studying the topological properties of almost periodic functions and demonstrate the validity of several known results, including some from [NGu84, BuNgu04] [[GaNGu07] to this general setting, as given in [KA11].

CHAPTER 12

Non-Archimedean Function Spaces Let X be a Hausdorff topological space. As mentioned earlier, the strict topology was introduced for the first time by Buck in [4] on the space of all bounded continuous functions on a locally compact space X. Several other authors have extended Buck’s results by taking as X an arbitrary completely regular space and considering spaces of continuous functions on X which are either real-valued or have values in a classical locally convex space or even in a classical topological vector space. In the non-Archimedean case some authors studied strict topologies on spaces of continuous functions, on a zero-dimensional topological space, with values either in a non-Archimedean valued field F or in a non-Archimedean locally convex space over F. In this chapter we will consider the case where the functions take their values in a non-Archimedean TVS. Throughout this chapter, unless it is stated explicitly otherwise, X will be a non-empty Hausdorff topological space, F a non-Archimedian nontrivially valued field (with unit element e) and E a non-trivial Hausdorff TVS over F with a base W of neighborhoods of 0 in E consisting of balanced and absorbing sets. As before, we denote by C(X, E) the space of all continuous E-valued functions on X. By Cb (X, E) (resp. Co (X, E), Coo (X, E), Crc (X, E)), we will denote the space of all f ∈ C(X, E) which are bounded (resp. vanish at infinity, vanish outside some compact set, have relatively compact range) in E. In case E = F, we will write C(X, F), Cb (X, F), Co (X, F), Coo (X, F) and Crc (X, F), respectively.

205

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1. Compact-open Topology on C(X, E) In this section, we shall study the τK topology which is the nonArchimedian analogues of the usual compact open topology studied in earlier chapters. First we recall some terminology (§ A.1, A.5). A topological space (X, τ ) is called zero-dimensional if, for each x ∈ X and each neighborhood U of x, there exists a clopen (i.e., both closed and open) set V such that x ∈ V and V ⊆ U. The family of all clopen subsets of X will be denoted by Clo(X). A valued field F is said to be non-trivially valued if there is an α ∈ F such that 0 < |α| = 6 1. By the F-characteristic function of a set A ⊆ X, we mean the function χA : X → F defined by χA (x) = e if x ∈ A and χA (x) = 0 if x ∈ / X (x ∈ X). Note that, for any ϕ ∈ C(X, F) and ε > 0, the set {t ∈ X : |ϕ(t)| ≥ ε} is clopen in X since it is the inverse image of the clopen set F\B(0, ε) under a continuous map. For any non-zero scalar λ ∈ F , we shall denote its inverse by λ−1 (instead of λ1 ). By S(X, F) we will denote the space spanned by the functions χA , A ∈ Clo(X), while S(X, F) ⊗ E denotes the space spanned by the functions χA ⊗ a, A ∈ Clo(X) and a ∈ E. Let (E, τ ) be a topological vector space (in short, a TVS) over a valued field (F, | · |). A subset S of E is called a non-Archimedean set if S + S ⊆ S. The pair (E, τ ) is called a non-Archimedean TVS if it has a base of τ neighborhoods of 0 consisting of non-Archimedean sets . Every non-Archimedean neighborhood of 0 in a TVS is a clopen (i.e. both closed and open) set. Every non-Archimedean valued field (F, | · |) is a non-Archimedean TVS over itself. Further, any indiscrete TVS is a non-Archimedean TVS. In particular, the trivial TVS E = {0} is a non-Archimedean TVS. Recall that a subset A of X is said to be bounding if every continuous real valued function on X is bounded on A. If every f ∈ C(X, F) is bounded on A, then A is said to be F-bounding. It is easy to see that continuous images of bounding (resp. F-bounding ) sets are bounding (resp. F-bounding ). The space X is said to be pseudocompact (resp. F-pseudocompact), if every continuous real function on X (resp. every f ∈ C(X, F)) is bounded. Lemma 12.1.1 If X is zero-dimensional, then a subset A of X is bounding iff it is F-bounding. Proof. Assume that A is not bounding and let g be a continuous real function on X which is not bounded on A. Then there exist a sequence (nk ) of positive integers and a sequence (xk ) in A such that nk < |g(xk )|
1, and let h = ∞ k=1 λ χBk .Then h ∈ C(X, F) and h(xk ) = λ , and so h is not bounded on A, which implies that A is not F-bounding. Now the result clearly follows. Recall that a completely regular Hausdorff space Y is c-complete or universally complete iff Y is homeomorphic to a closed subspace of a product of metrizable topological spaces. If Y is c-complete, then every bounding subset of Y is relatively compact. The c-completion of Y is the ˇ smallest c-complete subspace of the Stone-Cech compactification βY of Y , which contains Y , and it is denoted by θY . Definition. A basic sequence in E is a sequence U = (Vn ) of balanced absorbing subsets of E such that (1) Vn+1 + Vn+1 ⊆ Vn for all n. (2) For each n and each non-zero scalar λ, there exists an m such that Vm ⊆ λVn . Every basic sequence U = (Vn ) induces a pseudometrizable linear topology τU on E for which the sequence (Vn ) is a base at zero. If every Vn ∈ W, then the basic U is called topological. Given any such U, the set kerU = ∩Vn is a subspace of E. Let EU = E/ ker U be the quotient space and π = πU the quotient map. Then π(U) = (π(Vn )) is a basic sequence in EU . The following Theorem is contained in [BBNW75]. Theorem 12.1.2. Every Hausdorff topological vector space E over K is topologically isomorphic to a subspace of a product of metrizable topological vector spaces. Proof. We will only sketch the proof. Let F be the collection of all topological basic sequences in E and let G = ΠU ∈F EU . Then the map ϕ : E → G given by ϕ(a)U = πU (a), a ∈ E, is a linear homeomorphism between E and ϕ(E).

Remark. Using the preceding Theorem we get that E is completely regular.

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12. NON-ARCHIMEDEAN FUNCTION SPACES ˆ E

If Eˆ is the completion of E, then the family of all closures W in ˆ ˆ Using this fact, we get the E, for closed W ∈ W, is a base at 0 in E. following Lemma 12.1.3. If A ⊆ E, then A is totally bounded in E iff it is ˆ totally bounded in E. Proof. It is clear that if A is totally bounded in E, then it is totally ˆ For the converse, suppose that A is totally bounded in Eˆ bounded in E. and let W ∈ W be a closed. Choose another closed W1 ∈ W such that ˆ and W1 + W1 ⊆ W . Let V and V1 be the closures of W and W1 in E ˆ let S = {e1 , e2 . . . , en ) be a finite subset of E such that A ⊆ S + V1 . For k = 1, . . . , n, choose ak ∈ E such that ek − ak ∈ V1 . Given x ∈ A, there exist k and z ∈ V1 such that x = ek + z ∈ ak + V1 + V1 ⊆ ak + V. Thus x − ak ∈ V ∩ E = W and so A ⊆ {a1 , a2 . . . , an } + W .

Theorem 12.1.4 (1). If E is complete, then E is c-complete. (2). Every bounding subset of E is totally bounded. (3). If E is complete, then for a subset A of E the following are equivalent: (a) A is bounding. (b) A is totally bounded. (c) A is relatively compact. Proof. (1). By Theorem 12.1.2, E is topologically isomorphic to a subspace F of a product Πi∈I Ei of metrizable topological vector spaces. The space F is closed in the product since E is complete. It follows that E is c-complete. (2). Let A be a bounding subset of E. Then A is a bounding subset of ˆ E the completion Eˆ of E. Since Eˆ is c-complete, the closure A is compact and hence totally bounded, which implies that A is totally bounded. (3). (a)⇒ (b) follows from (2). (b)⇒(c) follows since E is complete. (c)⇒(a) Suppose A is relatively compact. Then A is compact and hence bounding, which implies that A is bounding. Theorem 12.1.5 Assume that X is zero-dimensional. For a subset A of X, the following are equivalent: (1) A is bounding. (2) A is F-bounding. (3) For each f ∈ C(X, E), the set f (A) is totally bounded in E. (4) For each f ∈ C(X, E), the set f (A) is bounded in E.

1. COMPACT-OPEN TOPOLOGY ON C(X, E)

209

Proof. (1)⇔(2) follows from Lemma 12.1.1. (1)⇒(3) follows since continuous images of bounding sets are bounding and in view of the preceding Theorem 12.1.4. (3)⇒(4) is clear. (4)⇒(1) Assume that (4) holds and let ϕ ∈ C(X, F). Since E is Hausdorff and non-trivial, there exists an u ∈ E and a closed balanced W ∈ W not containing u. Since the function f = ϕ ⊗ e is bounded on A, there exists a non-zero scalar λ such that ϕ(A)u ⊆ λW . If now x ∈ A, then λ−1 ϕ(x)u ∈ W and so |λ−1 ϕ(x)| < 1 since W is balanced and u ∈ / W . It follows that |ϕ(x)| < |λ| for all x ∈ A. Thus A is F-bounding. Notation. For V ∈ W and A ⊆ X, we denote by N(A, V ) : = N(χA , V ) = {f ∈ C(X, E) : f (A) ⊆ V }, Nb (A, V ) : = Nb (χA , V ) = N(A, V ) ∩ Cb (X, E). Throughout, K denotes a family of compact subsets of X which covers X and it is upwards directed by set inclusion. If A ∈ K is compact, then each N(A, V ) is absorbing. Further, if V + V ⊆ W , then N(A, V ) + N(A, V ) ⊆ N(A, W ). We get now easily the following Theorem 12.1.6. The family ΩK,B = {N(A, V ) : A ∈ K, V ∈ W } is a base at 0 in C(X, E) for a Hausdorff linear topology τK . Moreover τK does not depend on the particular choice of B. Note. (1) Let Ao ∈ K and Vo ∈ W. If (Vn ) is a basic topological sequence in E such that V1 +V1 ⊆ Vo , then (N(Ao , Vn )) is a basic sequence in C(X, E) and N(Ao , V1 ) + N(Ao , V1 ) ⊆ N(Ao , Vo ). (2) If K = K(X) is the family of all finite(resp. compact) subsets of X, we will denote τK by τk (resp. τρ ). (3) The topology of uniform convergence τu on Cb (X, E) has a base of neighborhood of 0 consisting of all sets of the form N(X, V ), V ∈ W . Recall that a sequence (Bn ), of subsets of of a TVS G, is called a fundamental sequence of bounded sets, if each Bn is bounded and each bounded subset of G is contained in some Bn . Lemma 12.1.7 If (C(X, E), τK ) has a fundamental sequence (Gn ) of bounded sets, then E also has the same property. Proof. For each A ∈ K and each n the set Gn (A) is bounded in E. In particular, since K covers X, each Gn (x), x ∈ X, is bounded in E. For a ∈ E, denote by a¯ the function which is defined on X by a ¯(x) = a.

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Let Bn = {a ∈ E : a ¯ ∈ Gn }. For x ∈ X, we have that Bn ⊆ Gn (x) and so Bn is a bounded subset of E. Let B be any bounded subset of E and G = {¯a : a ∈ B}. Since G(A) ⊆ B for every A ∈ K, it follows that G is a bounded subset of (C(X, E), τK ) and so G ⊆ Gn , for some n, which implies that B ⊆ Bn . Therefore (Bn ) is a fundamental sequence of bounded subsets of E. Recall that a Hausdorff zero-dimensional topological space X is a P -space if every countable intersection of clopen subsets of X is clopen. Theorem 12.1.8. If X is a zero-dimensional P -space and E sequentially complete, then each of the spaces (C(X, E), τρ ) and (C(X, E), τk ) is sequentially complete. Proof. (1). Let {fn } be a τρ -Cauchy sequence. Since E is sequentially complete, the limit f (x) = lim fn (x) exists in E for each x. The function f is continuous. Indeed, let x ∈ X and let V ∈ W be closed. Let V1 be the interior of V and let m be such that fn (x) ∈ f (x) + V1 for all n ≥ m. For each n ≥ m, there exists a clopen T neighborhood An of x such that fn (An ) ⊆ f (x) + V1 . The set A = n≥m An is clopen and fn (y) ∈ f (x) + V for all y ∈ A and all n ≥ m.

It follows that f (y) ∈ f (x) + V for all y ∈ A, which proves that f is continuous at x. (2). Suppose that {fn } is τk -Cauchy. Then {fn } is τρ -Cauchy and hence fn → f pointwise, for some f ∈ C(X, E). Given a closed V ∈ W and a compact subset Y of X, there exists no such that (fn − fm )(Y ) ⊆ V for all n, m ≥ no . Since V is closed and lim fm (x) = f (y) for all y, it follows that (fn − τk f. f )(Y ) ⊆ V for all n ≥ no , which proves that fn → Theorem 12.1.9. Let K be a compact subset of the zero-dimensional topological space X and let g be a continuous E-valued function on K. Then: (1) Given a V ∈ W, there exists an f ∈ S(X, F) ⊗ E such that f (X) is a finite subset of g(K) and (g − f )(K) ⊆ V . (2) If E is complete and metrizable, then there exists a continuous E-valued function f on X which is an extension of g and whose image in E is relatively compact.


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Proof. (1). For each x ∈ K, there exists a clopen neighborhood Ax of x in K such that g(y) − g(x) ∈ V for all y ∈ Ax . By the compactness of K, there are x1 , x2 , . . . , xn in K such that K = ∪nk=1 Axk . Let n−1 Bk . B1 = Ax1 , B2 = Ax2 \ B1 , . . . , Bn = Axn \ ∪k=1

For each k there exists a clopen subset Zk of X such that Zk ∩ K = Bk . Replacing Zk by Zk \ ∪j6=k Zj , we may assume that P the sets Z1 , . . . , Zn are pairwise disjoint. Now it suffices to take f = nk=1 χZk g(xk ). (2). Assume that E is metrizable and complete and let λ ∈ F, 0 < |λ| < 1. There exists a base (Vn ) of closed balanced neighborhoods of 0 in E such that Vn+1 + Vn+1 ⊆ Vn and Vn+1 ⊆ λVn for all n. By (1), there exists a function f1 ∈ S(X, F) ⊗ E such that f1 (X) ⊆ g(K) and (g − g1 )(K) ⊆ V2 , where g1 = f1 |K . Applying (1) to the function g − g1 , we get a function f2 ∈ S(X, F) ⊗ E such that f2 (X) ⊆ (g − g1 (K) and (g − g1 − g2 )(K) ⊆ V3 , where g2 = f2 |K . Continuing by induction, we get a sequence {fn } ∈ S(X, F) ⊗ E such that, for gk = fk |K , we have that (g −

n X

gk )(K) ⊆ Vn+1 and fn (X) ⊆ (g −

n−1 X

gk )(K) for all n.

k=1

k=1

P∞ Claim I The series n=1 fn converges Pn uniformly to a continuous function f on X. Indeed, let hn = k=1 fk and let V ∈ W closed There existsPan m such that Vm ⊆ V . For m < n < N, we have that hN − hn = N k=n+1 fk . But, for k > n, we have ! k−1 X fk (X) ⊆ g − gi (K) ⊆ Vk i=1

and so

(hN − hn )(X) ⊆ Vn+1 + Vn+2 + . . . + VN ⊆ Vn ⊆ V. Thus the sequence {hn } is τu -Cauchy in Cb (X, E) and hence it converges uniformly to a continuous function f ∈ Cb (X, E). It is easy to see that f (X) is totally bounded in E and hence relatively compact.

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Claim II f = g on K. Indeed, let y ∈ K and let V ∈ W be closed and balanced. There exists an m such that Vm ⊆ V . Let n ≥ m. Then n X g(y) − fk (y) ∈ Vn+1 ⊆ V. k=1

Since V is closed, we get that g(y) − f (y) ∈ V . This, being true for all closed balanced neighborhoods V of 0 in E, implies that g(y) = f (y) since E is Hausdorff. Hence the result follows. Corollary 12.1.10. If X is zero-dimensional, then S(X, F) ⊗ E is τk -dense in C(X, E). Theorem 12.1.11. Assume that X is zero-dimensional and let K be an upwards directed family of compact subsets of X covering X. Then (C(X, E), τK ) is complete iff E is complete and every f ∈ F (X, E) such that f |A is continuous, for each A ∈ K, is continuous on X. Proof. (⇒) Suppose that (C(X, E), τK ) is complete. Since K covers X, it follows easily that E is complete. Let f ∈ F (X, E) be such that f |A is continuous, for each A ∈ K. Consider the the set F of all pairs (A, V ), where A ∈ K and V ∈ W closed and balanced. We make F into a directed set by defining (A1 , V1 ) ≥ (A2 , V2 ) iff A2 ⊆ A1 and V1 ⊆ V2 . Given α = (A, V ) ∈ F , there exists (by Theorem 12.1.9) a function gα = gA,V in C(X, E) such that (f − gα )(A) ⊆ V . The net (gα )α∈F is Cauchy in (C(X, E), τK ). Indeed, let V ∈ W be balanced and Ao ∈ K. Choose a closed balanced Vo ∈ W such that Vo + Vo ⊆ V . Let αo = (Ao , Vo ) and let αi = (Ai , Vi ), i = 1, 2, αi ≥ αo . Then (f − gαi )(Ao ) ⊆ Vi ⊆ Vo and so (gα1 − gα2 )(Ao ) ⊆ Vo + Vo ⊆ V. This proves that the net (gα )α∈F is Cauchy in (C(X, E), τK ) and hence it converges to some g ∈ C(X, E). We will show that g = f . Indeed, let xo ∈ X and let V ∈ W. Choose a closed balanced Vo ∈ W such that Vo + Vo ⊆ V . Let Ao ∈ K containing xo . There exist A1 ∈ K containing Ao and a closed balanced V1 ∈ W contained in Vo such that (g − gα )(Ao ) ⊆ Vo for all α ≥ α1 = (A1 , V1 ). Thus g(xo ) − gα1 (xo ) ⊆ (g − gα1 )(Ao ) ⊆ Vo . Also f (xo ) − gα1 (xo ) ⊆ (f − gα1 )(A1 ) ⊆ V1 ⊆ Vo


213

and hence g(xo ) − f (xo ) ∈ Vo − Vo ⊆ V . It follows that f = g and so f is continuous. (⇐) Suppose that E is complete and every f ∈ F (X, E) is continuous if f |A is continuous for all A ∈ K. Let (fα ) be a Cauchy net in (C(X, E), τK ). For each x ∈ X, the net {fα (x)} is Cauchy in E and hence convergent. Define f (x) = lim fα (x). If V ∈ W is a closed balanced and Ao ∈ K, there exists αo such that (fα1 − fα2 )(Ao ) ⊆ V for all α1 , α2 ≥ αo . Thus (fα − f )(Ao ) ⊆ V for all α ≥ αo , which proves that fα → f uniformly on Ao . This implies that f |A is continuous for each A ∈ K and hence f is continuous, Moreover, fα → f in (C(X, E), τK ). Corollary 12.1.12. Suppose that X is zero-dimensional. Then (C(X, E), τρ ) is complete iff E is complete and X discrete. Proof. (⇒) In view of the preceding Theorem, if E is complete and X discrete, then (C(X, E), τρ ) is complete. (⇐) Conversely, suppose that (C(X, E), τρ ) is complete. Then E is complete. Let A ⊆ X and a a non-zero element of E. Let f = χA a. For every finite subset B of X, f |B is continuous and hence f is continuous by the Theorem 12.1.11. The set A is closed. Indeed let x ∈ A¯ \ A and let {xα } be a net in A converging to x. Then f (xα ) → f (x) = 0. There exists an open V ∈ W not containing a. Since f (x) = 0 ∈ V , there exists a α such that a = f (xα ) ∈ V , a contradiction. Thus every subset of X is closed and hence X is discrete.

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2. Strict Topology on Cb (X, E) In this section, we shall study the τP topology which is the nonArchimedian analogues of the usual strict topology studied in earlier chapters. For ϕ ∈ F (X, F) and V ∈ W, we set N(ϕ, V ) = {f ∈ Cb (X, E) : (ϕf )(X) ⊆ V }. It is easy to see that: (1) If ϕ ∈ B(X, F), then N(ϕ, V ) is absorbing in Cb (X, E). (2) For λ a non-zero scalar, we have γN(ϕ, V ) = N(λ−1 ϕ, V ) = N(ϕ, λV ). (3) If V is balanced, then N(ϕ, V ) is balanced. (4) If V ⊆ V1 ∩ V2 and |ϕ| ≥ max{|ϕ1 |, |ϕ2|} and if V is balanced, then \ N(ϕ, V ) ⊆ N(ϕ1 , V1 ) N(ϕ2 , V2 ).

(5) If V is convex, then N(ϕ, V ) is convex. Let now P be an upwards directed family of compact subsets of X covering X. We denote by BoP (X, F) the collection of all bounded functions ϕ ∈ F (X, F) which P-vanish at infinity, i.e. for each ε > 0 there exists an A ∈ P such that |ϕ(x)| < ε if x ∈ / A. We denote by βP the linear topology on Cb (X, E) for which the family of all N(ϕ, V ), ϕ ∈ BoP (X, F) and V ∈ W, is a base at 0. It follows easily that βP has as a base at 0 the family of all N(ϕ, V ), with V ∈ W and ϕ ∈ BoP (X, F), kϕk ≤ 1. In case P is the family of all compact (resp. all finite) subsets of X, we will denote the corresponding βP by βo (resp. βs ). Let τu denote the topology of uniform convergence on Cb (X, E). Theorem 12.2.1. (1) τρ 6 τP 6 βP 6 τu on Cb (X, E). (2) τu , βP and βs have the same bounded sets. (3) τP and βP coincide on τu -bounded sets. (4) A sequence {fn } in Cb (X, E) is βP -convergent to some f iff it is τP f. τu -bounded and fn → (5) If (Cb (X, E), βP ) is bornological, then βP = τu . (6) If X is zero-dimensional, then (a) τu = βP iff X ∈ P . (b) τP = βP iff, for each sequence (An ) in P , there exists A in P which contains every An . (c) Coo (X, E) is βo -dense in Cb (X, E) iff X is locally compact. Proof. (1). Since P covers X and is upwards-directed, it follows easily that τρ is coarser than τP . Also τP is coarser than βP since the

2. STRICT TOPOLOGY ON Cb (X, E)

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F-characteristic function of any A ∈ P belongs to BoP (X, F). Finally, let V ∈ W be balanced and ϕ ∈ P, kϕk ≤ 1. Then {f ∈ Cb (X, E) : f (X) ⊆ V } ⊆ N(ϕ, V ), which proves that βP is coarser than τu . (2). Assume that there exists a subset G of Cb (X, E) which is βs bounded but not τu -bounded. Then there exists a balanced V ∈ W such that G is not absorbed by the set W = {f ∈ Cb (X, E) : f (X) ⊆ V }. Let λ ∈ F, |λ| > 1. Suppose that we have already chosen f1 , f2 , . . . , fn in G and distinct x1 , x2S , . . . , xn in X such that fk (xk ) ∈ / λ2k V , for k = n 1, . . . , n. The set D = k=1 fk (X) is bounded in E. Hence there exists |γ| > |λ|2(n+1) such that D ⊆ γV and G ⊆ γ · Nb (Z, V ) where Z = {x1 , x2 , . . . , xn } and Nb (Z, V ) = {f ∈ Cb (X, E) : f (Z) ⊆ V }. Since G is not absorbed by Nb (X, V ), there exists fn+1 in G which is not in γNb (X, V ) and hence γ −1 fn+1 (xn+1 ) ∈ / V , for some xn+1 . But then xn+1 ∈ / Z. Thus, we get by induction a sequence {fn } in G and a sequence {xn } of distinct elements of X such that fn (xn ) ∈ / λ2n V for all n. Now define ϕ ∈ F (X, F) by ϕ(xn ) = λ−n and ϕ(x) = 0 if x ∈ / {x1 , x2 , . . .}. If S is the family of all finite subsets of X, then ϕ ∈ BoS (X, F) and so G ⊆ αN(ϕ, V ) for some non-zero scalar α. Let n be such that |λ|n > |α|. Now α−1 λ−n fn (xn ) = α−1 fn (xn )ϕ(xn ) ∈ V, and so fn (xn ) ∈ αλn V ⊆ λ2n V , a contradiction. This clearly completes the proof of (2). (3). Let G be a τu -bounded set, fo ∈ G, ϕ ∈ BoP (X, F), with kϕk ≤ 1, and V ∈ W be balanced. There exists a balanced V1 ∈ W with V1 + V1 ⊆ V . Since G is τu -bounded, there exists a non-zero scalar λ such that G(X) ⊆ λV1 . Choose A ∈ P such that |ϕ| < |λ|−1 on X \ A. Now \ [fo + Nb (A, V1 )] G ⊆ fo + N(ϕ, V ). Indeed, let f ∈ G be such that f − fo ∈ Nb (A, V1 ). If x ∈ A, then f (x) − fo (x) ∈ V1 and so ϕ(x)[f (x) − fo (x)] ∈ V1 ⊆ V . If x ∈ / A, then −1 |ϕ(x)| < |λ| and hence ϕ(x)[f (x) − fo (x)] ∈ λ−1 λV1 − λ−1 λV1 ⊆ V. Thus f ∈ fo + N(ϕ, V ), which clearly proves that τP = βP on G.

216


(4) (⇒) If {fn } is βP -convergent, then it is βP -bounded and hence τu -bounded. Also {fn } is clearly τP -convergent. (⇐)This follows from (3). (5) It follows from the fact that βP is coarser than τu and the topologies βP and τu have the same bounded sets. (6). Suppose that X is zero-dimensional. (a). (⇒) If X ∈ P, then the constant function ϕ(x) = 1, for all x ∈ X, is in BoP (X, F), and {f ∈ Cb (X, E) : f (X) ⊆ V } = N(ϕ, V ). Thus τu is coarser than βP and so τu = βP . (⇐) Assume that τu = βP and let V ∈ W be a proper balanced set (such a neighborhood exists since E is Hausdorff and non-trivial). Choose a balanced V1 ∈ W and ϕ ∈ BoP (X, F), 0 < kϕk ≤ 1, such that N(ϕ, V1 ) ⊆ Nb (X, V ). Let a ∈ E \ V and choose a non-zero scalar λ such that a ∈ λV1 . There exists A ∈ P such that |ϕ(x)| < |λ|−1 if x ∈ / A. Now A = X. Indeed, suppose that there exists an x ∈ X \ A. Since X is zero-dimensional, there exists a clopen neighborhood Z of x disjoint from A. If f = χZ a, then f ∈ N(ϕ, V1 ) but f ∈ / Nb (X, V ), a contradiction. (b) (⇒)Suppose thatPβP = τP and let 0 < |λ| < 1. Let (An ) be a n sequence in P and ϕ = ∞ n=1 λ χAn . Then ϕ ∈ BoP (X, F). Let V ∈ W be proper and balanced and a ∈ E \ V . There exists a balanced V1 ∈ W and A ∈ P such that Nb (A, V1 ) ⊆ N(ϕ, V ). Now assume that there exists an x ∈ (∪An ) \ A and let n be the smallest integer such that x ∈ An . Then |ϕ(x)| = |λ|n . Choose a clopen neighborhood Z of x disjoint from A and let f = λ−n χZ a. Clearly f ∈ Nb (A,SV1 ) but f ∈ / N(ϕ, V ) since |ϕ(x)λ−n | = 1 and a ∈ / V . This proves that An ⊆ A. (⇐) Assume that the condition is satisfied and let V ∈ W be balanced and ϕ ∈ BoP (X, F), with kϕk ≤ 1. Let 0 < |λ| < 1 and choose, for each positive integer n, an An ∈ P such that |ϕ(x)| < |λ|n when x ∈ / An . By our hypothesis, there exists an A ∈ P containing each An . Now Nb (A, V ) ⊆ N(ϕ, V ). In fact, let f ∈ Nb (A, V ) and x ∈ X. If ϕ(x) 6= 0, then |ϕ(x)| ≥ |λ|n , for some n, and so x ∈ An ⊆ A, which implies that f (x) ∈ V and thus ϕ(x)f (x) ∈ V . (c) (⇒) Assume that X is locally compact and let V ∈ W be balanced, ϕ ∈ BoP (X, F), with kϕk ≤ 1, and f ∈ Cb (X, E). There exists a non-zero scalar λ such that f (X) ⊆ λV and an A ∈ P such that |ϕ(x)| < |λ|−1 when x ∈ / A. Every y ∈ A has a compact clopen neighborhood. By the compactness of A, there exists a compact clopen set Z containing A. If g = χZ f , then g ∈ Coo (X, E). Moreover, if g(y) 6= f (y), then g(y) = 0 and y ∈ / A, which implies that −ϕ(y)f (y) ∈ −λϕ(y)V ⊆ V , since V is


217

balanced. It follows that g − f ∈ N(ϕ, V ), which proves that Coo (X, E) is βP -dense in Cb (X, E). (⇐) Assume that some x ∈ X has no compact neighborhood. If f ∈ Coo (X, E), then the set F = {y : f (y) 6= 0} is open. Also F is contained in a compact set B ∈ P. Thus x ∈ / F and so f (x) = 0. Choose any non-zero element a of E and let V ∈ W be balanced not containing a. The set D = Nb (x, V ) is a τρ -neighborhood of 0 and hence a βP -neighborhood of zero. Let f ∈ Cb (X, E), f (y) = a for all y. If g ∈ Coo (X, E), then g(x) − f (x) = −f (x) ∈ / V and so g − f ∈ / D, which implies that f is not in the βP -closure of Coo (X, E). Hence the result follows. Theorem 12.2.2. Assume that X is zero-dimensional and let G = (Cb (X, E), βP ). Then G is complete iff E is complete and every bounded f ∈ F (X, E) such that f |A is continuous, for each A ∈ P, is continuous on X. Proof. (⇒) Suppose that G is complete. It is easy to see that E is complete. Let f ∈ F (X, E) be bounded and such that f |A is continuous , for each A ∈ P. Consider the family ∆ of all pairs (A, V ), where A ∈ P and V ∈ W is balanced. We make ∆ into a directed set by defining (A1 , V1 ≥ (A2 , V2 ) iff A2 ⊆ A1 and V1 ⊆ V2 . For each α = (A, V ) in ∆, there exists a gα ∈ S(X, F) ⊗ E such that gα (X) ⊆ f (A) and (f − gα )(A) ⊆ V. The net {gα } is τu -bounded. Also the net is τP -Cauchy. Indeed, let Ao ∈ P and V ∈ W be balanced. Choose a closed balanced Vo ∈ W such that Vo + Vo ⊆ V . If αi = (Ai , Vi ) ≥ αo = (Ao , Vo ), i = 1, 2, then (f − gαi )(Ao ) ⊆ (f − gαi )(Ai ) ⊆ Vi ⊆ V and so (gα1 − gα2 )(Ao ) ⊆ Vo + Vo ⊆ V which proves that {gα } is τP Cauchy. Let D = {g ∈ Cb (X, E) : g(X) ⊆ f (X)}. Then F = D − D is uniformly bounded. Let now N be a βP -neighborhood of zero. There exists a τP -neighborhood N1 of 0 such that N ∩ F = N1 ∩ F since βP coincides with τP on F . There exists a αo such that gα − gα′ ∈ N1 if α, α′ ≥ αo . For such α, α′ ≥ αo we have that gα − gα′ ∈ N . This proves that the net {gα } is βP -Cauchy and it converges to some g ∈ Cb (X, E). Now g(x) = lim gα (x) for each x ∈ X. But lim gα (x) = f (x). Indeed, Let V ∈ W be balanced and choose another one Vo ∈ W which is closed and such that Vo + Vo ⊆ V . Let xo ∈ X and choose Ao ∈ P containing xo . Since {gα } converges to g with respect to the topology τP , there exists

218


a α1 = (A1 , V1 ), where Ao ⊆ A1 and V1 ⊆ Vo , such that, for α ≥ α1 , we have (g − gα )(Ao ) ⊆ Vo . Thus, g(xo ) − gα1 (xo ) ∈ (g − gα )(Ao ) ⊆ Vo . Also f (xo ) − gα1 (xo ) ∈ Vo and so g(xo ) − f (xo ) ∈ Vo + Vo ⊆ V . Thus f (xo ) = g(xo ) since E is Hausdorff. Thus f = g is continuous. (⇐) Suppose that E is complete and the condition is satisfied. Let (fγ ) be a Cauchy net in G. For each x ∈ X, the net (fγ (x)) is Cauchy in E and hence convergent. Define f (x) = lim fγ (x). For each ϕ ∈ BoP (X, F), the function ϕf is bounded. In fact, let V ∈ W and choose another one V1 ∈ W, which is closed and balanced, such that V1 + V1 ⊆ V . There exists a γo such that ϕ(fγ − fγ ′ )(X) ⊆ V1 if γ, γ ′ ≥ γo . Since V1 is closed, we get that ϕ(fγo − f )(X) ⊆ V1 . Since ϕ and fγo are bounded, there exists |λ| ≥ 1 such that (ϕfγo )(X) ⊆ λV1 . It follows that (ϕf )(X) ⊆ V1 + λV1 ⊆ λV, which proves that ϕf is bounded. Next we show that f is bounded. Assume the contrary. Then, there exists a balanced V ∈ W which does not absorb f (X). Hence, there are 0 < |λ1 | < |λ2 | < . . ., with lim |λn | → ∞, and a sequence (xn ) such that f (xn ) ∈ λ2n+1 V \ λ2n V for all n. Define ϕ on X by ϕ(xn ) = λ−1 / {x1 , x2 , . . .}. Then n and ϕ(x) = 0 if x ∈ ϕ ∈ BoP (X, F) but ϕf is not bounded since ϕ(xn )f (xn ∈ / λn V . This contradiction proves that f is bounded. Next we prove that f is continuous. In fact, let V ∈ W be closed balanced and A ∈ P. We get easily that there exists γo such that (fγ − f )(A) ⊆ V for all γ ≥ γo. Thus fγ (x) → f (x) uniformly on A and so f |A is continuous. This, by our hypothesis, implies that f is continuous on X. Moreover fγ → f in G. Theorem 12.2.3. If X is zero-dimensional, then (Cb (X, E), βP ) is metrizable iff E is metrizable and X ∈ P . Proof. This follows from Theorem 12.2.1 and the fact that a Hausdorff TVS over F is metrizable iff it has a countable base at 0 [BBNW75] (see also Section A.2).


219

Theorem 12.2.4. βP has as a base at 0 the family of all sets N(λn , An , V ) of the form N(λn , An , V ) =

∞ \

{f ∈ Cb (X, E) : f (An ) ⊆ λn V },

n=1

where 0 < |λn | → ∞, (An ) a sequence in P and V ∈ W. Proof. (⇒) Let N = N(λn , An , V ) be as in the Theorem. We may assume that V is balanced. Also we may assume that (An ) is increasing since P is upwards directed. As 0 < |λn | → ∞, for each n there exists a scalar γn such that |γn | = min{|λk | : k ≥ n}. Now |γn | → ∞. Let n1 < n2 , . . . be such that |γ1 | = . . . = |γn1 | < |γn1 +1 | = . . . = |γn2 | < |γn2 +1 | = . . . and consider N1 = N(γnk , Ank , V ). Then N1 ⊆ N .. [Indeed, let f ∈ N(γnk , Ank , V ) and set no = 0. For each positive integer k, there exists a j such that nj−1 < k ≤ nj . If x ∈ Ak , then x ∈ Anj and so f (x) ∈ γk V ⊆ λk V since |γk | = |γnj |. Thus N1 ⊆ N .] Let αk = γnk and Bk = Ank . The P −1 function ϕ = ∞ k=1 αk χBk belongs to BoP (X, F). Moreover N(ϕ, V ) ⊆ W . Indeed, let f ∈ N(ϕ, V ) and let x ∈ Anj = Bj . Let k be the smallest positive integer with x ∈ Bk . Then |ϕ(x)| = |αk |−1 . Now ϕ(x)f (x) ∈ V . Since |ϕ(x)| ≥ |αj |−1 , we have that f (x) ∈ αj V . This proves that N(ϕ, V ) ⊆ N(γnk , Ank , V ) ⊆ N(λn , An , V ) and so N(λn , An , V ) is a βP neighborhood of zero. (⇐) Let No be any βP -neighborhood of zero. There exist a balanced neighborhood V ∈ W and ϕ ∈ BoP (X, F), kϕk < 1, such that N(ϕ, V ) ⊆ Wo . Let 0 < |λ| < 1. For each n, there exists An ∈ P such that |ϕ(x)| < |λ|n if x ∈ / An . Set N2 = N(λ−n+1 , An , V ). We will finish the proof by showing that N2 ⊆ N(ϕ, V ). So let f ∈ N2 . If ϕ(x) 6= 0, then there exists an n such that |λ|n < |ϕ(x)| ≤ |λ|n−1 . Now x ∈ An and so f (x) ∈ λ−n+1 V , which proves that ϕ(x)f (x) ∈ V since |ϕ(x)λ−n+1 | ≤ 1. Theorem 12.2.5. If N is a βP -neighborhood of zero, then there exists a balanced V ∈ W such that, for each non-zero scalar λ, there exist A ∈ P and γ ∈ K, γ 6= 0, such that {f ∈ Cb (X, E) : f (X) ⊆ λV, f (A) ⊆ γV } ⊆ N . Proof. By Theorem 12.2.4, there exists a balanced V ∈ W, an increasing sequence (An ) in P and a sequence (λn ) in F, with 0 < |λn | → ∞, such that N1 = N(λn , An , V ) ⊆ N . Let now λ be a non-zero scalar

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and choose n such that |λk | > |λ| when k > n. If γ ∈ F is such that |γ| = min{|λ1 |, |λ2|, . . . , |λn |}, then {f ∈ Cb )(X, E) : f (X) ⊆ λV, f (An ) ⊆ γV } ⊆ N . Theorem 12.2.6. Let X, Y be Hausdorff topological spaces and let h : X → Y be a continuous function. Then the induced linear map T = Th : Cb (Y, E) → Cb (X, E),

f 7→ f ◦ h,

is βo − βo continuous. Proof. Let N be a βo -neighborhood of 0 in Cb (X, E). In view of Theorem 12.2.4, there exist a sequence (λn ) of scalars, where 0 < |λn | → ∞, a sequence (An ) of compact subsets subsets of X and V ∈ W such that N1 = N(λn , An , V ) ⊆ N . Each Bn = h(An ) is compact in Y . Moreover, N2 =

∞ \

{f ∈ Cb (Y, E) : f (Bn ) ⊆ λn V } ⊆ T −1 (N1 )

n=1

and so T is continuous. Theorem 12.2.7. If X is a zero-dimensional P -space and E sequentially complete, then (Cb (X, E), βo ) is sequentially complete. Proof. Let {fn } be a βo -Cauchy sequence in Cb (X, E). Then {fn } is βo -bounded and hence τu -bounded. Also, {fn } is τk -Cauchy and hence τk f . We show first that f is there exists f ∈ C(X, E) such that fn → bounded. Assume the contrary and let |λ| > 1. There exist a closed balanced V ∈ W and a sequence {xn } of distinct elements of X such that f (xn ) ∈ / λn V for all n. Choose a closed balanced V1 ∈ W such that V1 + V1 ⊆ V . The function ϕ(xn ) = λ−n and ϕ(x) = 0 if x ∈ / {xn : n = 1, 2, . . .} belongs to Bo (X, F). Hence there exists no such that [ϕ(fn − fm )](X) ⊆ V1 for all n, m ≥ no , and so [ϕ(fn − f )](X) ⊆ V1 for all n ≥ no . Since {fn } is uniformly bounded, there exists k ≥ no such that fn (X) ⊆ λk V1 for all n. Now f (xk ) = (f (xk ) − fk (xk )) + fk (xk ) ∈ λk V1 + λk V1 ⊆ λk V, a contradiction. Thus f is bounded. Since τk coincides with βo on τu βo τk f , we get that fn → f . bounded sets and since fn →


221

Theorem 12.2.8. Let X be zero-dimensional and let G be an S(X, F)-submodule of Cb (X, E). Then G is βP -dense in Cb (X, E) iff , for each x ∈ X, the set G(x) = {f (x) : f ∈ G} is dense in E. Proof. (⇒) Assume that G is βP -dense and let x ∈ X. Consider the linear map x b : Cb (X, E) → E given by x b(f ) = f (x), f ∈ Cb (X, E),

which is clearly βP -continuous. Thus

¯ ⊆x E=x b(Cb (X, E)) = x b(G) b(G) = G(x)

and so G(x) is dense in E. (⇐) Suppose that the condition is satisfied and let N = N(ϕ, V ), where V ∈ W is open and balanced and ϕ ∈ BoP (X, F). Given f ∈ Cb (X, E), there exists a non-zero scalar λ such that f (X) ⊆ λV and kϕk ≤ |λ|. Let K ∈ P be such that |ϕ(x)| < |λ|−1 if x ∈ / K. For each −1 x ∈ K, choose gx ∈ G such that gx (x) − f (x) ∈ λ V . Since X is zero dimensional, there exists a clopen neighborhood Zx of x such that gx (y) − f (y) ∈ λ−1 V for all y ∈ Zx . By the compactness of K, there are x1 , x2 . . . , xn ∈ K such that K ⊆ S n k=1 Zxk . Let D1 = Zx1 and Dk = Zxk \

k−1 [

Dj for k = 2, 3, . . . , n.

j=1

Pn Each hk = χDk gxk belongs to G and so g = k=1 hk ∈ G. Now g − f ∈ S N(ϕ, V ). Indeed if x ∈ / nk=1 Dk , then g(x) = 0 and |ϕ(x)| < |λ|−1 which implies that ϕ(x)[g)(x) − f (x)] = −ϕ(x)f (x) ∈ V because V is balanced. For x ∈ Dk we have ϕ(x)[g(x) − f (x)] = ϕ(x)[gxk (x) − f (x)] ∈ V .

Corollary 12.2.9. If X is zero-dimensional, then S(X, F) ⊗ E and Crc (X, E) are βP -dense in Cb (X, E).

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3. Maximal Ideals in NA Function Algebras In this section, we give a characterizations of maximal and β-closed ideals in the algebra Cb (X, E) (k-closed ideals in the algebra C(X, E)) when E is a topological algebra. Recall that the strict topology β on Cb (X, E) is the linear topology which has a base of neighborhoods of 0 in Cb (X, E) consisting of all sets of the form N(ϕ, W ) = {f ∈ Cb (X, E) : (ϕf )(X) ⊆ W }, where ϕ ∈ Bo (X, F) and W ∈ W. It is shown in ([Kat84a], Proposition 2.5) that, in case X is locally compact and 0-dimensional, the above definition of strict topology is equivalent to the one given in ([Pro77], p. 198-199), where ϕ is assumed to vary over Co (X, F) instead of Bo (X, F). We first consider a version of the Stone-Werierstrass theorem for Cb (X, F)-submodules in Cb (X, E). Theorem 12.3.1. [KKK11] Assume that either X or E is 0-dimensional (in particular, E is a non-Archimedean TVS), and let A be a vector subspace of Cb (X, E) which is a Cb (X, F)-module. If, for each x ∈ X, A(x) is dense in E, then A is β-dense in Cb (X, E). Proof. Let f ∈ Cb (X, E), and let ϕ ∈ Bo (X, F) and W an open balanced neighborhood of 0 in E. Choose λ ∈ F with ||ϕ|| < |λ| and f (X) ⊆ λW . There exists a compact set K of X such that |ϕ(x)| < |λ|−1 for x ∈ / K. For each x ∈ K, there is a gx ∈ A such that f (x) − gx (x) ∈ λ−1 W. Let G(x) = {y ∈ X : f (y) − gx (y) ∈ λ−1 W } = (f − gx )−1 (λ−1 W ). If E is 0-dimensional, we may assume that W is clopen, and therefore G(x) is also clopen in X. If X is 0-dimensional, there exists clopen neighborhood N(x) of x such that N(X) ⊆ G(x). We see that in any case, there exists a clopen neighborhood Vx of x such that Vx ⊆ G(x) = {y ∈ X : f (y) − gx (y) ∈ λ−1 W }. By the compactness of K, there are x1 , x2 , ..., xn in Y such that K ⊆ ∪nk=1 Vxk . Let D1 = Vx1 and Dxk+1 = Vxk+1 \ ∪ni=1 Vxi . The sets D1 , ..., Dn are pairwise disjoint and clopen. P Let ϕk be the Fcharacteristic function of Dk . The function g = nk=1 ϕk gxk belongs to

3. MAXIMAL IDEALS IN NA FUNCTION ALGEBRAS

223

A. Moreover, [ϕ(f − g)](X) ⊆ W . [Indeed, let x ∈ X. If x ∈ Dk , then g(x) = gxk (x) and so ϕ(x)[f (x) − g(x)] ∈ ϕ(x)λ−1 W ⊆ W. If x ∈ / ∪nk=1 Dk , then g(x) = 0 and x ∈ / Y . Thus ϕ(x)[f (x) − g(x)] = ϕ(x)f (x) ∈ ϕ(x)λW ⊆ W.] β

Thus f ∈ A . A useful equivalent form of the above is the following theorem which characterizes β-closed submodules in Cb (X, E). Theorem 12.3.2. [KKK11] Assume that either X or E is 0dimensional (in particular, E is a non-Archimedean TVS), and let A be a vector subspace of Cb (X, E) which is a Cb (X, F)-module. Then, for β any f ∈ Cb (X, E), f ∈ A iff f (x) ∈ A(x) for all x ∈ X. In particular, A is β-closed iff, for each x ∈ X, f (x) ∈ A(x) for all f ∈ A. β

Proof. Assume that f ∈ A , and let x ∈ X. If ϕ is the characteristic function of {x}, then ϕ ∈ Bo (X, F) and so there exists a g ∈ A such that: f (x) − g(x) ∈ [ϕ(f − g)](x) ⊆ W , which implies that f (x) ∈ A(x). The converse part follows from Theorem 12.3.1. Recall that if E is an associative algebra over a valued field F and τ a topology on E, the pair (E, τ ) is called a topological algebra if (E, τ ) is a TVS and the multiplication (x, y) → xy from E × E into E is jointly continuous (i.e. for any τ neighborhood W of 0 in E, there exists a τ neighborhood H of 0 in E such that H 2 ⊆ W, where H 2 = {xy : x, y ∈ H}). (E, τ ) is called topological simple if E has no two-sided closed ideals other than {0} and E. It is easy to verify that if (E, τ ) is a topological algebra, then (Cb (X, E), β) is also a topological algebra. Theorem 12.3.3. [KKK11] Let E be a topological algebra with identity over F. Assume that either X or E is 0-dimensional (in particular, E is non-Archimedean) Let Ω be the set of all pairs α = (N, W ), where N is a non-empty closed subset of X and W a proper closed two-sided ideal in E. For αi = (Ni , Wi ), i = 1, 2, define α1 6 α2 iff N2 ⊆ N1 and W1 ⊆ W2 . Also let Jα = S(N, W ) = {f ∈ Cb (X, E) : f (N) ⊆ W } for α = (N, W ). Then: (a) Each Jα is a proper two-sided ideal in Cb (X, E) which is ρ-closed and hence β-closed.

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(b) Jα1 ⊆ Jα2 iff α1 6 α2 . Thus the map α → Jα is one-to-one and order preserving. (c) A β-closed two-sided ideal J in Cb (X, E) is maximal iff there exist a unique xo ∈ X and a unique maximal closed two-sided ideal Wo in E such that J = S(xo , Wo ). (d) If E is topologically simple and J is a β-closed two-sided ideal in Cb (X, E), then: (i) There exists a closed subset N of X such that J = I(N) = {f ∈ Cb (X, E) : f (N) = {0}}. (ii) J is maximal iff there exists a (unique) x ∈ X such that J = I(x). Proof. (a) It is easy to see that Jα is a two-sided ideal in Cb (X, E) which is a-closed since W is closed. Moreover Jα is a proper subset of Cb (X, E). Indeed let a ∈ E\W . Then the constant function f (x) = a is not in Jα . (b) It is clear that Jα1 ⊆ Jα2 when α1 6 α2 . Conversely, suppose that Jα1 ⊆ Jα2 . If x ∈ N2 \N1 , then there exists a clopen neighborhood V of x which is disjoint from N1 . Let a ∈ E\W2 and consider the function g = χA ⊗ a, where χA is the F-characteristic function of V . Then g ∈ Jα1 \Jα2 , a contradiction. Thus N2 ⊆ N1 . Also if b ∈ W1 \W2 , then the constant function g(x) = b is in Jα1 \Jα2 , a contradiction. Thus W1 ⊆W2 and (b) follows. (c) We first note that J is a Cb (X, F)-module. Indeed, let f ∈ J and h ∈ Cb (X, F). If u is the identity of E, then the function g = h⊗u belongs to Cb (X, E) and so gf ∈ J . Since gf = hf , our claim follows. Now β suppose that J is maximal and let f ∈ Cb (X, E)\J . Since f ∈ /J =J , there exists, by Theorem 12.3.2, xo ∈ X such that f (xo ) ∈ / J (xo ) = Wo . Since Wo is a proper ideal and J ⊆ S(xo , Wo ), we have that J = S(xo , Wo ). The ideal Wo is maximal. [Indeed, let W1 be a proper closed two-sided ideal which contains Wo . Then J ⊆ S(xo , W1 ) 6= Cb (X, E), and so S(xo , Wo ) = S(xo , W1 ), which implies that W1 = Wo .] The uniqueness of xo and Wo follows from (b). Conversely, suppose that J = J(xo ,Wo) , where Wo is a maximal closed ideal in E and let J1 be a proper closed two-sided ideal in Cb (X, E) β which contains J . Let f ∈ Cb (X, E)\J1 . Since f ∈ / J1 = J1 , there exists, by Theorem 12.3.2, y ∈ X such that f (y) ∈ / J1 (y) = W . Now J(xo ,Wo ) ⊆ J1 ⊆ J(y,W ) . But then Wo ⊆W and y = xo (by (b)). Since Wo

3. MAXIMAL IDEALS IN NA FUNCTION ALGEBRAS

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is maximal, we have that Wo = W and so J =J1 = J(y,W ) , which proves that J is maximal. (d) Suppose that E is topological simple. Then: (i) Let J be a β-closed two-sided ideal in Cb (X, E). Then J is a Cb (X, F)-module. Define N = {x ∈ X : f (x) = 0 for all f ∈ J }. Then N is closed in X. Further, J ⊆ I(N). We now show that I(N) ⊆ J . [Let g ∈ I(N), but g ∈ / J . Since J is β-closed, by Theorem 12.3.2, β

J = J = {f ∈ Cb (X, E) : f (x) ∈ J (x) for all x ∈ X}; so there exists an xo ∈ X such that g(xo ) ∈ / J (xo ). Therefore J (xo ) 6= E. Since E is simple and J (xo ) is a two-sided ideal in E, we have J (xo ) = {0}. In particular, f (xo ) = 0 for all f ∈ J and so xo ∈ N. But g(xo) ∈ / J (xo ) and so g(xo ) 6= 0 (as 0 ∈ J (xo )). This is a contradiction, since g ∈ I(N) and xo ∈ N imply that g(xo ) = 0. Therefore I(N) ⊆ J .] Thus J = I(N). (ii) It follows from (c) since the only maximal closed ideal in E is {0}. Remark. Some results related to the above ones for the ”compactopen topology” on C(X, E), due to Prolla ([Pro82], p. 218-222), may be stated without proof as follows: Theorem 12.3.4. [Pro82] Assume that either X or E is 0-dimensional (in particular, E is a non-Archimedean TVS), and let A be a vector subspace of C(X, E) which is a C(X, F)-module. Then, for any f ∈ k C(X, E), f ∈ A iff f (x) ∈ A(x), for all x ∈ X. Theorem 12.3.5. [Pro82] Let E be a topological algebra with identity over F. Assume that either X or E is 0-dimensional (in particular, E is non-Archimedean). Then: (a) For any non-empty closed subset N of X and a proper closed two-sided ideal W in E, S(N, W ) is a proper two-sided ideal in C(X, E) which is k-closed. (b) A k-closed two-sided ideal J in C(X, E) is maximal iff there exist a unique xo ∈ X and a unique maximal closed two-sided ideal Wo in E such that J = S(xo , Wo ) = {f ∈ C(X, E) : f (xo ) ∈ Wo }. (c) If E is simple and J is a k-closed two-sided ideal in C(X, E), then:

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(i) There exists a closed subset N of X such that J = I(N) = {f ∈ C(X, E) : f (N) = {0}}. (ii) J is maximal iff there exists a (unique) x ∈ X such that J = I(x).


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4. Notes and Comments Section 12.1. In this section, a generalized form of compact-open topology τK is considered on the non-Archimedean function space C(X, E). Earlier authors assumed X a zero-dimensional topological space and E a non-Archimedean valued field F or a non-Archimedean LCS over F. In fact, the first result on the Weierstrass type approximation theorem in the non-Archimedean area was proved by Dieudonne [Dieu44] in 1944 for C(X, F), where F is a p-adic field. This result was extended in 1950 by I. Kaplansky [Kap50] to the case of F a non-Archimedean valued field (i.e. a rank one valued field). These results were completed by Chernoff, Rasala and Waterhouse [CRW68] in 1968 who extended them to any field F with a krull valuation (or Archimedean valuations other than the usual absolute value of C). When E is a non-Archimedean normed space, Prolla [Pro77, Pro78] obtained a Stone-Weierstrass theorem which was later extended by Carneiro [Car79] to weighted function spaces CV (X, E). In his work, Carneiro considered locally F-convex spaces E over a local field (F, | · |), i.e. over a locally compact valued field (F, | · |). Later, Soares [So80 ], Prolla [Pro82] and Prolla and Verdoodt [PV97] obtained further generalizations where the field (F, |·|) is not assumed to be locally compact. Here we have considered the case where the functions take their values in a TVS over a non-Archimedian field F. The main results deal with completeness, completion, extension of function and denseness of subspaces, as given in [Kat11]. Section 12.2. This section makes a parallel study of the strict topology βP on Cb (X, E) assuming X a zero-dimensional topological space and E a non-Archimedean TVS, similar to that in Section 12.1, as given in [Kat11]. As mentioned earlier, the strict topology was introduced for the first time by Buck in 1958 on the space of all bounded continuous functions on a locally compact space X. Several other authors have extended Buck’s results by taking as X an arbitrary completely regular space and considering spaces of continuous functions on X which are either real-valued or have values in a classical locally convex space or even in a classical topological vector space. In the non-Archimedean case some authors studied strict topologies on spaces of continuous functions, on a zerodimensional topological space, with values either in a non-Archimedean valued field F or in a non-Archimedean locally convex space over F. In

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this paper we will consider the case where the functions take their values in a non-Archimedean linear topological space. Section 12.3. In this section, we first consider a Stone-Weierstrass type theorem in the non-Archimedean setting for the spaces (Cb (X, E), β) and (C(X, E), k), as given in [KKK11, Pro82].]. As applications, we give a characterizations of maximal and closed ideals in the algebra when E is a topological algebra.

APPENDIX A

Topology and Functional Analysis

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A. TOPOLOGY AND FUNCTIONAL ANALYSIS

1. Topological Spaces In this section we present some basic definitions and results on topological spaces. Definition. A topology on a non-empty set X is a collection τ of subsets of X having the following properties: (i) X and the empty set ∅ belong to τ . (ii) The union of an arbitrary number of sets in τ belongs to τ . (iii) The intersection of a finite number of sets in τ belongs to τ . In this case (X, τ ) is called a topological space and each set U in τ is called an open set. A subset A of X is said to be closed set if its complement X\A is an open set. The closure of a set A in X, denoted by A (or cl(A)), is defined as the intersection of all closed set containing A (i.e., A is the smallest closed set containing A). The interior of a set A in X, denoted by A0 (or int(A)), is defined as the union of all open set contained in A (i.e., A0 is the largest open set contained in A). Further, A is open iff A = A0 ; A is closed iff A = A. A is said to be dense in X if A = X. Definition. Let (X, τ ) be a topological space. (1) For any x ∈ X, a subset V of X is called a neighborhood of x if there exists an open set W such that x ∈ W ⊆ V. In particular, every open set U containing x is called an open neighborhood of x. The collection Ux of all neighborhoods of x is called the neighborhood system at x. (2) A base of neighborhoods of x (or a local base) at x is a subcollection Bx of Ux such that, for each U ∈ Ux , there exists some V ∈ Bx such that V ⊆ U; in this case, the elements of Bx are called basic neighborhoods of x. Theorem A.1.1. Let (X, τ ) be a topological space. For any x ∈ X, let Bx be a base of neighborhoods of x. Then: (NB1 ) x ∈ U for all U ∈ Bx . (NB2 ) If U, V ∈ Ux , there is some W ∈ Bx such that W ⊆ U ∩ V . (NB3 ) If V ∈ Bx , there is some Vo ∈ Bx such that if y ∈ Vo , there is some W ∈ By with W ⊆ V (i.e. V contains a basic neighborhood of each y ∈ Vo ), Furthermore, (NB 4 ) G ⊆ X is open iff G contains a basic neighborhood of each of its points.

1. TOPOLOGICAL SPACES

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Conversely, given a set X and that for each x ∈ X is given a nonempty collection Bx of subsets of X satisfying (NB 1 )−(NB 3 ) and if we define ”open” using (NB 4 ), there exists a unique topology µ on X such that Bx is a base of µneighborhoods of x for each x ∈ X. Definition. Let (X, τ ) be a topological space. (1) A subcollection B ⊆ τ is called a base ( or basis) for τ if, for any U ∈ τ and x ∈ U, there exists V ∈ B such that x ∈ V ⊆ U (or equivalently each U ∈ τ is a union of sets in B). (2) A subcollection S ⊆ τ is called a subbase (or subbasis) for τ if the finite intersection of sets in S form a base for τ . Remark. Any collection B of subsets of a set X is a base for some topology on X iff (i) X = ∪{B : B ∈ B}, and (ii) for any B1 , B2 ∈ B and x ∈ B1 ∩ B2 , there is some B3 ∈ B with x ∈ B3 ⊆ B1 ∩ B2 . On the other hand, any collection of subsets of a set X is a subbase for some topology on X. In fact, this is the smallest topology containing the given collection of sets. Definition. (1) If X is a non empty set, then a function d : X ×X → R is called a pseudometric on X if, for any x, y, z ∈ X, (M1 ) d(x, y) ≥ 0; (M2 ) if x = y, then d(x, y) = 0; (M3 ) d(x, y) = d(y, x) (symmetry); (M4 ) d(x, y) ≤ d(x, z) + d(z, y) (triangle inequality). If, in addition, d satisfies: d(x, y) = 0 ⇒ x = y, then d is called a metric on X. If d is a pseudometric (resp. metric) on a set X, then the pair (X, d) is called a pseudometric (resp. metric) space. For any x ∈ X and r > 0, the sets B(x, r) = {y ∈ X : d(y, x) < r} and B[x, r] = {y ∈ X : d(y, x) ≤ r} are called open ball and closed ball, respectively, with center x and radius r. (2) A subset U of a pseudometic space (X, d) is called a d-open set if, for each x ∈ U, there exists r > 0 such that B(x, r) ⊆ U. The collection td of all d-open sets is a topology on X, called the pseudometric topology induced by d. Clearly, the collection B = {B(x, r) : x ∈ X, r > 0} is a base for the topology td . (3) A topological space (X, τ ) is said to be pseudometrizable (resp. metrizable) if there exists a pseudometric (resp. metric) d on X such that τ = td .

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Definition. Let (X, d) be a metric space. (1) A sequence {xn : n ∈ N} is said to be convergent to x ∈ X if, given any ε > 0, there exists an integer no such that d(xn , x) < ε for all n ≥ no ; in this case, we write xn → x. (2) {xn } is said to be a Cauchy sequence if, given any ε > 0, there exists an integer no such that d(xn , xm ) < ε for all n, m ≥ no . Clearly, every convergent sequence is a Cauchy sequence. Definition. Let (X, d) be a metric space, and let A ⊆ X. Then A is called complete if every Cauchy sequence in A converges to a point in A. Definition. Let (X, τ ) be a topological space and A ⊆ X. (1) A point x in A is called an interior point of A if there exists an open set U in X such that x ∈ U ⊆ A. (2) A point x ∈ A is called an isolated point of A if it has a neighborhood which does not contain any other point of A. (3) A point x ∈ X is called a limit point of A if each neighborhood of x contains a point of A other than x. The set of all limit points of A is denoted by Ad . Note that, for any A ⊆ X, its closure A = A ∪ Ad ; so A is closed iff every limit point of A is in A. Further, every x ∈ A is either an isolated point of A or a limit point of A. Definition. (1) A relation ≥ on a set D is called a direction (or a directed relation) if: (i) α ≥ α,for all α ∈ D (reflexive), (ii) if α ≥ β, β ≥ γ, then α ≥ γ (transitive); (iii) if α, β ∈ D, there exists a γ ∈ D such that γ ≥ α, γ ≥ β (directive). In this case, (D, ≥) is called a directed set. (2) A net in a set X is a mapping f of a directed set D into X. We denote this net by {f (α) : α ∈ D} or {xα : α ∈ D}, where xα = f (α) ∈ X. If B is a subset of a directed set (D, ≥), with the induced ordering is directed, then (B, ≥) is called a directed subset of (D, ≥). A subnet {xβ : β ∈ B} in X is a directed subset of a net {xα : α ∈ D}. (3) A net {xα : α ∈ D} in a topological space (X, τ ) is said to be convergent to x ∈ X if, given any neighborhood U of x, there exists


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αo ∈ D such that xα ∈ U for all α ≥ αo , (i.e., xα ∈ U eventually). In this case, x is called the limit of the net and we write xα −→ x. Theorem A.1.2. Let X be a topological space and A ⊆ X. Then: (a) For any x ∈ X, x ∈ A iff there exists a net {xα } ⊆ A such that xα −→ x. (b) A is closed iff, for any net {xα } ⊆ A with xα → x ∈ X, we have x ∈ A. Definition. Let X and Y be topological spaces. (1) A function f : X → Y is said to be continuous at xo ∈ X if, for each neighborhood Vo of f (xo ) in Y , there exists a neighborhood Uo of xo in X such that f (Uo ) ⊆ Vo . f is said to be continuous on X if it is continuous at each x ∈ X. (2) A function f : X → Y is called a homeomorphism if f is continuous, one-one, onto and f −1 : Y → X is continuous. It is often convenient to use the following equivalent conditions for continuity. Theorem A.1.3. Let X and Y be topological spaces and f : X → Y a function. (a) The following conditions are equivalent: (i) f : X → Y is continuous on X. (ii) For each open set V ⊆ Y, f −1 (V ) is open in X. (iii) For each closed set W ⊆ Y, f −1 (W ) is closed in X. (iv) For any net {xα } ⊆ X with xα → x in X, f (xα ) → f (x) in Y . (b) If X and Y are metric spaces, then f : X → Y is continuous on X iff, for any sequence {xn } ⊆ X with xn → x in X, f (xn ) → f (x) in Y. (c) If τ1 and τ2 are two topologies on a set X, then the identity map i : (X, τ1 ) → (X, τ2 ) is continuous iff τ2 ⊆ τ1 . Remark. Clearly, if X and Y are topological spaces, then any function f : X → Y is continuous on X in each of the following cases: (1) X has the discrete topology τ = P (X), the power set of X; (2) Y has the indiscrete topology τ = {X, ∅}. Definition. A function f : X −→ Y is called an open function (resp. a closed function) if, for any open set (resp. closed set) U in X, f (U) is open (resp. closed) in Y .

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In general, f is continuous < f is open < f is closed. However, if f : X −→ Y is one-one and onto, then f is open iff f is closed iff f −1 is continuous; hence f is a homeomorphism iff f is continuous and open iff f is continuous and closed. Theorem A.1.4. (Map Gluing theorem) Let X and Y be topological spaces, and let X = A ∩ B, where A and B are closed subsets of X. Let f : A → Y and g : B → Y be continuous functions such that f = g on A ∩ B. Then the function h : X → Y defined by f (x) if x ∈ A, h(x) = g(x) if x ∈ B, is continuous on X. Proof. Let W be a closed set in Y . Then f −1 (W ) is closed in A and hence also in X, since A is closed in X. Similarly, f −1 (W ) is closed both in B and X. Hence h−1 (W ) = f −1 (W ) ∩ g −1 (W ), and thus h−1 (W ) is closed in X. This implies that, h is continuous on X.

Definition. A mapping ϕ : (X, d) −→ (Y, ρ) is called an isometry of (X, d) into (Y, ρ) if ρ(ϕ(x1 ), ϕ(x2 )) = d(x1 , x2 ) for all x1 , x2 ∈ X. ( i.e. ϕ preserves distances). In this case, X and ϕ(X) are said to be isometric.Clearly, every isometry is one-one, continuous and an open mapping. Hence an isometry ϕ : (X, d) → (Y, ρ) is a homeomorphism ⇐⇒ ϕ is onto. However, a homeomorphism ϕ : (X, d) → (Y, ρ) need not be an isometry. Definition. If X, Y, Z are topological spaces, then a function f : X × Y → Z is called (i) separately continuous on X × Y if, given any (x, y) ∈ X × Y and any neighborhood G of f (x, y) in Z, there exist neighborhoods H of x in X and J of y in Y such that f (H × {y}) ⊆ G and f ({x} × J) ⊆ G. (ii) jointly continuous on X × Y if, given any (x, y) ∈ X × Y and any neighborhood G of f (x, y) in Z, there exist neighborhoods H of x in X and J of y in Y such that f (H × J) ⊆ G. Clearly, jointly continuity ⇒ separately continuity; the converse need not hold. Definition. A topological space (X, τ ) is called: (1) a T1 -space if, for any two points x 6= y in X, there exist two open sets U, V ⊆ X such that x ∈ U, y ∈ V and x ∈ / V, y ∈ / U;


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(2) Hausdorff if, for any two points x 6= y in X, there exist two open sets U, V ⊆ X such that x ∈ U, y ∈ V and U ∩ V = ∅; (3) regular if, for any closed set A in X and x ∈ X with x ∈ / A, there exist two open sets U, V ⊆ X such that x ∈ U, A ⊆ V and U ∩ V = ∅; (4) normal if for any two closed sets A, B ⊆ X with A∩B = ∅, there exist two open sets U, V ⊆ X such that A ⊆ U, B ⊆ V and U ∩ V = ∅; (5) completely regular if, for any closed set A ⊆ X and x ∈ X with x ∈ / A, there exists a continuous function f : X → [0, 1] such that f (x) = 0 and f (y) = 1 for all y ∈ A. Definition. Let X be a topological space, and let A ⊆ X. (1) A collection U = {Gα : α ∈ I} of open subsets of X is said to be an open cover of A if A ⊆ ∪α∈I Gα . If U and V are two covers of X, then V is said to be a refinement of U if each V ∈ V is contained in some U ∈ U. (2) A is called compact if every open cover of A has a finite subcover. (3) A is called countably compact if every countable open cover of A has a finite subcover. (4) A is called sequentially compact if every sequence in A has a convergent subsequence with limit in A. (5) A is called relatively compact if its closure A is compact. (6) If A is a subset of a pseudometric space (X, d), then A is called precompact (or totally bounded) if, for each ε > 0, there exists a finite subset D = {x1 , ..., xn } of X such that A⊆

n [

B(xi , ε).

i=1

Definition. A topological space (X, τ ) is called: (1) locally compact if each x ∈ X has a relative compact neighborhood; (2) pseudocompact if every continuous real-valued function on X is bounded; (3) σ-compact if it can be written as a countable union of compact sets; (4) Lindelöf if every open cover of A has a countable subcover; (5) hemicompact if X can be expressed as a countable union of compact sets Kn such that each compact subset of X is contained in some Kn ;

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(6) paracompact if every open cover U of X has refinement V which is locally finite (i.e. each x ∈ X has a neighborhood which intersects only a finite number of members of V); (7) a k-space if, for any U ⊆ X, U is closed in X whenever U ∩ K is closed in K for each compact K ⊆ X; (8) a kR -space if a function f : X → R is continuous on X whenever f | K is continuous for every compact subset K of X; (9) separable if it has a countable dense subset; (10) zero-dimensional (or 0-dimensional) if, for each x ∈ X and each neighborhood U of x, there exists a clopen (i.e., both closed and open) set V such that x ∈ V and V ⊆ U; (11) totally disconnected if, for any x ∈ X, Cx = {x} , where Cx is the component of x (i.e., the maximal connected subset of X containing x). (12) X is called extremely disconnected (or Stonian) if the closure of every open set is open. We now summarize some properties of the above spaces in the following theorem. Theorem A.1.5. (a) A topological space X is Hausdorff iff every convergent net in X has a unique limit. (b) A closed subset A of a compact space X is compact. (c) A compact subset A of a Hausdorff space X is closed. (d) Every compact Hausdorff space is normal. (e) A subset A of a topological space X is compact iff every net in A has a convergent subnet with limit in A. (f ) Every compact space and every sequentially compact space is countably compact and hence pseudocompact. (g) A pseudometric space is compact iff it is sequentially compact iff it is countably compact iff it is precompact and complete. (h) Every locally compact space and every first countable (in particular, metric) space is a k-space. (i) The continuous image of a compact set is compact. (j) If X is a k-space and Y any topological space, then a function f : X → Y is continuous on X iff f | K is continuous for every compact subset K of X; hence every k-space is a kR -space. (k) Every locally compact σ-compact Hausdorff space is hemicompact. (l) Every compact metric space is separable. (m) Every open subspace of a separable space is separable; an arbitrary subspace of a separable metric space is also separable. (n) The continuous image of a separable space is separable.


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(o) Every zero-dimensional T1 -space is totally disconnected and completely regular. (p) A locally compact Hausdorff is 0-dimensional iff it is totally disconnected. (q) Continuous image of a totally disconnected space need not be totally disconnected. Examples. (1) Q, P and discrete spaces are zero-dimensional and totally disconnected. (2) Q is not extremely disconnected. Note. Clearly, a finite subset A of a topological space X is always compact; but need not be closed. So it need not be relatively compact (e.g. if X is neither T1 nor regular). Example. Let X = N with the topology τ = {N} ∪ {An : n ∈ N}, where An = {1, 2, ...n − 1} for every n ∈ N. This is T0 but not T1 (hence also not Hausdorff). In this case, the only closed compact subsets of X and ∅, so the only relatively compact set is ∅. Theorem A.1.6. Let X be any topological space and Y a Hausdorff topological space, and let f, g : X → Y be continuous. Then: (a) The set A = {x ∈ X : f (x) = g(x)} is closed in X. (b) If D ⊆ X and f = g on D, then f = g on D. In particular, if D is dense in X and f = g on D, then f = g on X. Notation. For any function ϕ : X → R, A ⊆ X, and t ∈ R, we shall often write ϕ(A) = t to mean that ϕ(A) = {t}, that is, ϕ(x) = t for all x ∈ A. Theorem A.1.7. (Urysohn lemma ) Let X be a normal space, and let A and B be closed subsets of X with A ∩ B = ∅. Then there exists a continuous function ϕ : X → [0, 1] such that ϕ(A) = 0 and ϕ(B) = 1. Theorem A.1.8. (a) Let X be a completely regular space, K a compact subset of X, and B a closed of X with K ∩ B = ∅. Then there exists a continuous function ϕ : X → R such that ϕ(K) = 0 and ϕ(B) = 1. (b) Let X be a locally compact Hausdorff space, K a compact subset of X, and U a neighborhood of K. Then there exists a continuous function ϕ : X → [0, 1] such that ϕ(K) = 1 and the support of ϕ is

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compact and contained in U. (The support of ϕ is the closure of the set {x ∈ X : ϕ(x) 6= 0}). In particular, every normal T1 -space and every locally compact Hausdorff space is completely regular. Further, every 0-dimensional T1 -space is completely regular. Definition. A topological space X is called a Baire space if whenever X = ∪∞ n=1 An with each An closed, then, for at least one m, int(Am ) 6= ∅. Theorem A.1.9. (Baire) (a) A topological space X is a Baire space iff, for any countable family {Un } of open dense subsets in X, ∩∞ n=1 Un 6= ∅. (b) Every locally compact Hausdorff space is a Baire space. (c) Every complete metric space is a Baire space. Theorem A.1.10. (Tietze extension theorem) Let X be a normal space, and let A be a closed subset of X. Then, for any continuous function f : A → [0, 1] (or R), there exists a continuous function g : X → [0, 1] (or R) such that g = f on A. (Here g is called a continuous extension of f from A to X). Definition. Let {(Xα , τα ) : α ∈ I} be an indexed family of topological spaces. The product topology on the Cartesian product Π Xα is α∈I

defined as the topology τΠ obtained by taking as its base the collection of all sets of the form Π Uα , where each Uα is τα -open in Xα and Uα = Xα α∈I

except for a finite number of indices α’s. Remark. If {Xα : α ∈ I} is a family of compact spaces, then Π Xα αεI

with the product topology τΠ is a compact space (Tychonoff theorem). Definition. If (X, τ ) is a completely regular Hausdorff space, let C = Cb (X, R), the set of all continuous and bounded functions f : X → R. For each f ∈ C, define a closed bounded (hence compact) interval in R by If = [ inf f (x), sup f (x)]. x∈X

x∈X

Define a map e : X → Π If (with the product topology) by f ∈C

e(x) = (f (x))f ∈C , x ∈ X. ˇ Then βX = e(X) is a compact space, called the Stone-Cech compactification of X.


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ˇ ech) Let X be a completely regular Theorem A.1.11. (Stone-C ˇ Hausdorff space and βX its Stone-Cech compactification. Let Y be a compact Hausdorff space or Y = R. Then every continuous function f : X → Y has a unique continuous extension fb : βX → Y .

Definition. (1) A subset A of a topological space X is called a zero set [GJ60] if there exists a continuous function ϕ : X → R and that A = ϕ−1 (0) = {x ∈ X : ϕ(x) = 0}.

In this case, we write A = Z(ϕ). The complement of a zero set is called a cozero set. (2) A subset D of X is called a Gδ -set if it is a countable intersection of open sets; D is called an Fσ -set if it is a countable union of closed sets. (3) X is called a P -space ([GJ60], p. 62-63) if every zero set (or equivalently, every Gδ -set) in X is open. Clearly, the complement of a Gδ -set is an Fδ -set, and vice versa. For any function f : X → R, the set D of all points of discontinuity of f is an Fσ -set and the set of all points of continuity of f is a Gδ -set. It follows from ([GJ60], 4K(3)) that every compact subset of a P -space is finite. Definition. [GJ60, Nag65]) A subset A of a topological space X is said to have finite covering dimension if there exists a non-negative integer n such that for each finite open cover U of A there exists an open refinement V of U which is a cover of A and such that any point of A belongs to a most n + 1 member of V. The least such n is called the covering dimension of A. If no such finite n exists, then we say that A has an infinite covering dimension. Theorem A.1.12. (a) For each n ≥ 1, the covering dimensions of Rn is n. (b) Every compact subset of a P -space has finite covering dimension. (c) If a normal space X has finite covering dimensions, then so has ˇ its Stone-Cech compactification βX. Recall that, if X is a topological space, then a function ϕ : X → R is continuous at xo ∈ X if, for any ε > 0, there exists an open neighborhood N(xo ) of xo such that |ϕ(y) − ϕ(xo )| < ε for all y ∈ N(xo ), or equivalently, ϕ(xo ) − ε < ϕ(y) < ϕ(xo ) + ε for all y ∈ N(xo ).

(*)

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Definition. If X is a topological space, then a function ϕ : X → R is said to be (a) upper semicontinuous at xo ∈ X if, for any r ∈ R with ϕ(xo ) < r, there exists an open neighborhood N(xo ) of xo such that ϕ(y) < r for all y ∈ N(xo ); (b) lower semicontinuous at xo ∈ X if, for any r ∈ R with ϕ(xo ) > r, there exists an open neighborhood N(xo ) of xo such that ϕ(y) > r for all y ∈ N(xo ); (c) upper (resp. lower ) semicontinuous on X if it is upper (resp. lower) semicontinuous at each point of X, or equivalently, if, for each r ∈ R, the set {x ∈ X : ϕ(x) < r} (resp. {x ∈ X : ϕ(x) > r}) is open ([Will70], p. 49). Clearly, by (∗), a function ϕ : X → R is continuous at xo ∈ X iff it is both upper and lower semicontinuous at xo ∈ X. Theorem A.1.13. (a) A function ϕ : X → R is continuous iff it is both upper and lower semicontinuous. (b) The characteristic function χA of a subset A of X is upper (resp. lower) semicontinuous iff A is closed (resp. open). (c) Every upper (resp. lower) semicontinuous function assumes its supremum (resp. infimum) on a compact set. In particular, every nonnegative upper semicontinuous function on a compact set is bounded. Definition: Let X and Y be topological spaces and T : X → Y a mapping. Then: (a) The set G(T ) = {(x, T x) : x ∈ X} is called the graph of T in X × Y. (b) T is said to have a closed graph if G(T ) is a closed subset of X ×Y. (Here X × Y has the product topology.) The following theorem gives equivalent condition for a mapping T : X → Y to have a closed graph. Theorem A.1.14. Let X and Y be topological spaces and T : X ×Y a mapping. Then the following are equivalent (i) T has a closed graph. (ii) If {xα } is a net in X with xα → x ∈ X and T xα → y ∈ Y , then y = T (x) (iii) For each x ∈ X and y 6= T (x) in Y , there exist neighborhoods U of x in X and V of y in Y such that T (U) ∩ V = ∅.


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Proof. (i) ⇒ (ii). Let {xα = α ∈ I} ⊆ X with xα → x ∈ X and T xα → y ∈ Y . Then clearly (xα , T xα ) → (x, y) in X × Y (in the product topology). Since {(xα , T xα ) = α ∈ I} ⊆ G(T ) and, by (i), G(T ) is closed, we have (x, y) ∈ G(T ). Hence y = T x. (ii) ⇒ (i). Suppose (ii) holds. To show that G(T ) ⊆ G(T ), let (x, y) ∈ X × Y with (x, y) ∈ G(T ). Then there exists a net {(xα , T xα ) : α ∈ I} ⊆ G(T ) such that (xα , T xα ) → (x, y). This implies that xα → x and T xα → y. Hence, by hypothesis (ii), y = T x. So (x, y) = (x, T x) ∈ G(T ). Thus G(T ) = G(T ) and so T has a closed graph. The next result is concerned with the questions: Under what conditions continuity of T implies that T has a closed graph ? Conversely, under what conditions, T has a closed graph implies that T is continuous ? Theorem A.1.15. Let X and Y be topological spaces with Y Hausdorff, and let T : X → Y be a mapping. If T is continuous, then T has a closed graph. Proof. Suppose T is continuous. To show that T has a closed graph, let (x, y) ∈ X×Y with (x, y) ∈ G(T ). Then there exists a net {(xα , T xα ) : α ∈ I} ⊆ G(T ) such that (xα , T xα ) → (x, y). This implies that xα → x and T xα → y. Since T is continuous and xα → x, we have T xα → T x in Y . Since Y is Hausdorff, {T xα } has a unique limit and so y = T x. Hence (x, y) = (x, T x) ∈ G(T ). Thus G(T ) = G(T ); i.e. T has a closed graph. The converse of the above theorem is known as the ”closed graph theorem” and will be considered in Section A.3. Definition: Let E be a vector space over the field K (= R or C). Then a function k.k : E → R is called a norm on E if (N1 ) kxk ≥ 0 for all x ∈ E; (N2 ) kxk = 0 iff x = 0; (N3 ) kλxk = |λ| kxk for all x ∈ E and λ ∈ K; (N4 ) kx + yk ≤ kxk + kyk for all x, y ∈ E. Definition. (1) A vector space E with a norm k.k is called a normed vector space or, simply, a normed space. Clearly, every normed space (E, k.k) is a metric space with metric given by: d(x, y) = kx − yk,

x, y ∈ E.

(2) A normed space E is called a Banach space if it is complete with respect to the metric induced by the norm.

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Definition. Let A be an algebra over K (= R or C) which is also a normed space with norm || · ||. Then (A, || · ||) is called (a) a normed algebra if the norm || · || is submultiplicative, i.e. ||xy|| ≤ ||x||.||y|| for all x, y ∈ A; (b) a Banach algebra if A is a complete normed algebra.

2. TOPOLOGICAL VECTOR SPACES

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2. Topological Vector Spaces In the sequel, we consider all vector spaces over the field K (= R or C) with the usual absolute value. Notations. Let E be a vector space and A, B ⊆ E , x ∈ E, α, β ∈ K. We denote x ± A = {x ± a : a ∈ A}; A ± B = {a ± b : a ∈ A, b ∈ B}; αA = {αa : a ∈ A}; αA + βB = {αa + βb : a ∈ A, b ∈ B}. Clearly (α + β)A ⊆ αA + βA, but αA + βA ⊆ (α + β)A need not hold, in general. Definition. If E is a vector space and τ is a topology on E, then the pair (E, τ ) is called a topological vector space (TVS, in short) if the following conditions hold: (TVS1 ) The operation of addition (x, y) → x + y of E × E → E is jointly continuous; i.e., given any x, y ∈ E and any τ neighborhood U of x + y in E, there exist τ neighborhoods V of x and W of y in E such that V + W ⊆ U. (TVS2 ) The operation of scalar multiplication (λ, x) → λx of K × E → E is jointly continuous; i.e., given any x ∈ E and λ ∈ K and any τ neighborhood U of λx in E, there exist a τ neighborhood V of x in E and a neighborhood D of λ in K such that DV ⊆ U. Examples: (1) Any normed space (E, k.k) is a TVS since the conditions (TVS1 ) and (TVS2 ) follow, respectively, from (i) If x, y ∈ E and xn → x, yn → y, then xn + yn → x + y in E. (ii) If x ∈ E, λ ∈ K and λn → λ, xn → x, then λn xn → λx in E. (2) Any vector space E with the indiscrete topology τI = {E, ∅} is a TVS. (3) A vector space E with the discrete topology τD = P (E) is not a TVS unless E = {0} (since the scalar multiplication need not be continuous). Definition: Let E be a vector space over the field K (= R or C). A subset A of a vector space E is called: (i) absorbing if, for each x ∈ E, there exists a number r = r(x) > 0 such that x ∈ λA for all λ ∈ K with |λ| ≥ r; (or equivalently, there exists a number t = t(x) > 0 such that λx ∈ A for all λ ∈ K with |λ| ≤ t);

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(ii) balanced if λx ∈ A for all x ∈ A and λ ∈ K with |λ| ≤ 1; (iii) convex if tx + (1 − t)y ∈ A for all x, y ∈ A and 0 ≤ t ≤ 1. Examples. (1) Any vector subspace A of E is balanced and convex, but it need not be absorbing. (2) If (E, ||.||) is a normed vector space, then, for each r > 0, both the open and closed balls B(0, r) = {x ∈ E : ||x|| < r}, B[0, r] = {x ∈ E : ||x|| ≤ r} are absorbing, balanced and convex subset of E. Lemma.A.2.1. Let (E, τ ) be a TVS. (a) For any fixed a ∈ E, the translation map fa : E → E given by fa (x) = x + a = a + x, x ∈ E, is a homeomorphism of E into itself. (b) For any fixed λ (6= 0) ∈ K, the scalar multiplication map gλ : E → E given by gλ (x) = λx, x ∈ E, is also a homeomorphism of E into itself. Proof. It is easy to see that fa−1 = f−a and gλ−1 = g1/λ . (a) Clearly fa is one-one and onto. Since fa−1 = f−a , by (TVS1 ), both fa and fa−1 are continuous. Hence fa is a homeomorphism of E into itself. (b) Since λ 6= 0, it easily follows that gλ is one-one and onto. Since −1 gλ = g1/λ , by (TVS2 ), both gλ and gλ−1 are continuous. Hence gλ is also a homeomorphism of E into itself. Lemma A.2.2. Let (E, τ ) be a TVS, and let a ∈ E and λ(6= 0) ∈ K. (a) If U is an open (respectively, a closed) set in E, then a + U and λU are also open (respectively, closed) in E. (b) U is a neighborhood of 0 iff a + U is a neighborhood of a. (c) U is a neighborhood of 0 iff λU is a neighborhood of 0. (d) A collection W of subsets of E is a base of neighborhoods of 0 in E iff the collection a + W = {a + U : U ∈ W} is a base of neighborhoods of a in E. Proof. (a) By above Lemma, the translation map fa : E → E is a homeomorphism; in particular, fa is both an open and closed map. Hence, if U is an open (resp. a closed) set in E, then fa (U) = a+U is also open (resp. closed) in E. Next, the map gλ : E → E is a homeomorphism;


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hence, if U is an open (resp. a closed) set in E, then gλ (U) = λU is also open (resp. closed) in E. (b) & (c). These follow immediately from (a). (c) By (b), U is a neighborhood of 0 iff a + U is a neighborhood of a in E. Hence W is a base of neighborhoods of 0 in E iff a + W is a base of neighborhoods of a in E. The following theorem shows that every TVS has a base W of neighborhoods of 0 consisting of closed, balanced and absorbing sets. Theorem A.2.3. Let (E, τ ) be a TVS, and let W be a base of neighborhoods of 0 in a TVS E. Then: (A1 ) For every U, V ∈ W, there exists W ∈ W such that W ⊆ U ∩V . (A2 ) For each U ∈ W, there exists V ∈ W such that V + V ⊆ U. (A3 ) For each U ∈ W and λ(6= 0) ∈ K, λU ∈ W. (A4 ) Each U ∈ W is absorbing. (A5 ) For each U ∈ W, there exists a balanced W ∈ W such that W ⊆ U. (A6 ) For each U ∈ W, there exists a closed neighborhood W ∈ W such that W ⊆ U. Conversely, if W is a non-empty collection of subsets of E satisfy (A1 ) − (A5 ) then there exist a topology τ on E making (E, τ ) a TVS with a base W of τ -neighborhoods of 0. Proof. (A1 ) This follows from the property (NB2 ) of a local base of neighborhoods of a topological space. (A2 ) Let U ∈ W. By (TVS1 ), the mapping f : (x, y) → x + y is continuous at (x, y) = (0, 0) ∈ E × E. Since f (0, 0) = 0 + 0 = 0 and U is a neighborhood of 0, there exist neighborhoods V1 and V2 of 0 such that V1 + V2 ⊆ U. Put V = V1 ∩ V2 . Then V ∈ W with V + V ⊆ V1 + V2 ⊆ U. (A3 ) Let U ∈ W and λ(6= 0) ∈ K. Choose an open neighborhood V of 0 such that V ⊆ U. By (TVS2 ), the function gλ : E −→ E defined by gλ (x) = λx (x ∈ E) is a homeomorphism and hence an open map. Hence gλ (V ) = λV is an open set. Now λV ⊆ λU, and so λU is a neighborhood of 0. Thus λU ∈ W. (A4 ) Let U ∈ W, and let xo ∈ E. By (TVS2 ), the mapping hxo : K → E, defined by hxo (λ) = λxo (λ ∈ K) is continuous at λ = 0. So there exist t > 0 such that hxo (B[0, t]) ⊆ U, where B[0, t] = {µ ∈ K : |µ| ≤ t}. Hence µxo ∈ U for all |µ| ≤ t, and so U is absorbing. (A5 ) Let U ∈ W. The function g : (λ, x) → λx is continuous at (λ, x) = (0, 0) by (TVS2 ). So there exist a neighborhoods V of 0 in E and B[0, r] of 0 in K such that W = B[0, r]V ⊆ U. Clearly, W is a balanced. Further, W is a neighborhood of 0 (since rV ⊆ B[0, r]V = W and rV is

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a neighborhood of 0 by (A3 )). Thus, we have a balanced W ∈ W such that W ⊆ U. (A6 ) Let U ∈ W. By (A2 ) and (A5 ), we can choose a balanced V ∈ W such that V + V ⊆ U. We now show that V ⊆ U. [Let x ∈ V . Then, since x + V is a neighborhood of x, (x + V ) ∩ V 6= ∅. Choose y ∈ (x + V ) ∩ V. Then y = x + v1 = v2 , where v1, v2 ∈ V. So x = v2 − v1 ∈ V − V = V + V ⊆ U. This shows that V ⊆ U.] Thus W = V ∈ W is closed with W ⊆ U. Conversely, let V = {V ⊆ E : V contains some U ∈ W}. Clearly, W ⊆ V. We first show that, for each x ∈ E, Bx = x + V satisfies (NB1 ) − (NB3 ). (NB1 ) By (A4 ), each U ∈ W is absorbing and so 0 ∈ U; hence 0 ∈ V for all V ∈ V. Therefore x = x + 0 ∈ x + V for all V ∈ V. (NB2 ) Let x + V1 , x + V2 ∈ Bx , where V1 , V2 ∈ V. We show that (x + V1 ) ∩ (x + V2 ) ∈ x + V. Now (x + V1 ) ∩ (x + V2 ) = x + V1 ∩ V2 . Choose U1 , U2 ∈ W such that U1 ⊆ V1 , U2 ⊆ V2 . By (A1 ), there exists a U3 ∈ W such that U3 ⊆ U1 ∩ U2 . Hence x + U3 ⊆ x + V1 ∩ V2 . (NB3 ) Let x + V ∈ Bx , where V ∈ V. Choose U ∈ W with U ⊆ V . By (A2 ), there exists a W ∈ W such that W +W ⊆ U. Now, if y ∈ x+W, then y + W ∈ By = y + V and y + W ⊆ x + W + W ⊆ x + V. Now (NB1 )-(NB3 ) are satisfied and so, by a general result on base of neighborhoods in Topology (Theorem A.1.1), there exists a unique topology τ on E such that Bx = x + V forms a base of τ neighborhoods of x for each x ∈ E. Clearly, for any x ∈ E and U ∈ W, x + U is a τ neighborhood of x (since U ∈ W ⊆ V). Hence x + W is a base of τ neighborhoods of x for each x ∈ E. In particular, W is a base of τ neighborhoods of 0. It remains to verify (TVS1 ) and (TVS2 ). (TVS1 ) By (A2 ), it follows that the map (x, y) → x + y is continuous at (x, y) = (0, 0). Let xo , yo ∈ E, and let U ∈ W. Then xo + yo + U is a τ neighborhood of xo + yo . By (A2 ), there exists a V ∈ W such that V + V ⊆ U. Then xo + V is a τ neighborhood of xo , yo + V is a τ neighborhood of yo, and (xo + V ) + (xo + V ) = xo + yo + V + V ⊆ xo + yo + U. Hence the map (x, y) → x + y is continuous at (x, y) = (xo , yo ) ∈ E × E.


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(TVS2 ). Let λo ∈ K and xo ∈ E. We show that the map g : (λ, x) → λx is continuous at (λ, x) = (λo , xo ). Let W ∈ W, so that λo xo + W is a τ neighborhood of λo xo . Choose a balanced V ∈ W such that V +V ⊆ W (by (A2 ) and (A5 )). Since V is absorbing (by A4 ), there exists a r > 0 such that xo ∈ rV . By (A3 ), U = r|λor|+1 V ∈ W. We now show that B(λo , 1/r)(xo + U) ⊆ λo xo + W . Let λ ∈ B(λo , 1/r) and x ∈ xo + U. Then, since V is balanced and 1 + r|λo| 1 + |λo | = , r r = λ(x − xo ) + (λ − λo )xo r ∈ λ· V + (λ − λo )rV r|λo | + 1 ⊆ V + V ⊆ W.

|λ| ≤ |λ − λo | + |λo | < so λx − λo xo

Consequently, the map g : (λ, x) → λx is continuous at (λ, x) = (λo , xo ). Thus (E, τ ) is a TVS. We now summarize topological properties of a TVS. Theorem A.2.4. Let (E, τ ) be a TVS. Then: (a) (E, τ ) is regular. T (b) (E, τ ) is Hausdorff iff U = {0}, where W is a base of τ U ∈W

neighborhoods of 0. (c) (E, τ ) is a T1 -space iff (E, τ ) is Hausdorff iff (E, τ ) is completely regular. (d) (E, τ ) is locally compact iff (E, τ ) has a compact τ -neighborhood of 0 iff (E, τ ) is homeomorphic to Kn for some n ≥ 1 (i.e. (E, τ ) is finite dimensional). (e) (E, τ ) is compact iff E = {0}. (f) (E, τ ) is both connected and locally connected. If a vector space E is non-trivial (i.e. E 6= {0}) with the indiscrete topology τI = {E, ∅}, then (E, τI ) is a TVS which is not a T1 -space, hence also not a Hausdorff space. But it is regular and normal. Definition. A subset A of a TVS E is called: (a) bounded if, for any neighborhood W of 0 in E, there exists r > 0 such that A ⊆ λW for all λ ∈ K with |λ| ≥ r, or, equivalently, A ⊆ rW if W is balanced.

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(b) precompact (or totally bounded) if, for each neighborhood W of 0, there exists a finite subset D = {x1 , ..., xn } of E such that A ⊆ D + W = ∪ni=1 (xi + W ). Clearly, compactness ⇒ precompactness ⇒ boundedness; the reverse implications need not be true. Definition. A net {xα : α ∈ I} in a TVS E is said to be convergent to x ∈ E if, given any neighborhood U of 0 in E, there exists an αo ∈ I such that xα − x ∈ U for all α ≥ αo ; in this case, we write xα → x. {xα } is said to be a Cauchy net if, given any neighborhood U of 0 in E, there exists an αo ∈ I such that xα − xβ ∈ U for all α, β ≥ αo . Definition. Let E be a TVS, and let A ⊆ E. Then A is called: (i) complete if every Cauchy net in A converges to a point in A; (ii) sequentially complete if every Cauchy sequence in A converges to a point in A; (iii) quasi-complete (or boundedly complete) if every bounded Cauchy net in A converges to a point of E (i.e. if every bounded closed subset of E is complete). Clearly, completeness ⇒ quasi-completeness ⇒ sequential completeness; the reverse implications need not hold. Theorem A.2.5. A subset A of a TVS E is compact iff it is complete and precompact. b is called a Definition. Let E be a TVS. Then a complete TVS E b such completion of E if there exists a topological isomorphism ϕ : E → E b Every TVS has a unique Hausdorff completion that ϕ(E) is dense inE. ([KN63], p. 63-64). Definition. A TVS (E, τ ) is called: (i) a locally convex space (or a locally convex TVS ) if it has a base of convex neighborhoods of 0; (ii) metrizable if τ is induced by a metric d on E; (iii) an F -space if it is a complete metrizable TVS. In general, E is locally convex < E is metrizable.


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Examples. (1) (i) ([Rud91], p. 86) For 0 < p < 1, the sequence space ∞ X ℓp (N) = {x = {xn } ⊆ K : |xn |p < ∞}, n=1

the set of all p-summable sequences in K, is a metrizable TVS (in fact, an F -space) with metric given by ∞ X |xn − yn |p , x = {xn }, y = {yn } ∈ ℓp (N). dp (x, y) = n=1

but it is not locally convex. (ii) ([Rud91], p. 36-37) For 0 < p < 1,the space Z b Lp [a, b] = {f : [a, b] −→ K: |f (t)|p dt < ∞}, a

the set of all Lebesgue p-integrable functions, is also an F -space but not locally convex. (2) For X a topological space, let E = C(X) with the compact-open topology k, i.e. the topology which has a base W of kneighborhoods of 0 consisting of all sets of the form U(K, ε) = {f ∈ C(X) : sup |f (x)| ≤ ε}, x∈K

where K ⊆ X is compact and ε > 0. Then each U(K, ε) is absorbing, balanced and convex, and so (C(X), k) is locally convex. However, (C(X), k) is not metrizable. In fact, if X is a locally compact Hausdorff space, then (C(X), k) is metrizable iff X is hemicompact (i.e. X can be expressed as a countable union of compact sets Kn such that each compact subset of X is contained in some Kn ). Definition. Let E be a vector space over K. (1) A function q : E → R is called a seminorm on E if it satisfies (S1 ) q(x) ≥ 0 for all x ∈ E; (S2 ) q(x) = 0 if x = 0; (S3 ) q(λx) = |λ|q(x) for all x ∈ E and λ ∈ K (absolutely homogeneous); (S4 ) q(x + y) ≤ q(x) + q(y) for all x, y ∈ E (subadditive). (2) A function q : E → R is called a k-norm on E, where 0 < k ≤ 1, if it satisfies (K1 ) q(x) ≥ 0 for all x ∈ E; (K2 ) q(x) = 0 if and only if x = 0; (K3 ) q(λx) = |λ|k q(x) for all x ∈ E and λ ∈ K;

250


(K4 ) q(x + y) ≤ q(x) + q(y) for all x, y ∈ E. (3) A function q : E → R is called an F -seminorm on E if it satisfies (F1 ) q(x) ≥ 0 for all x ∈ E; (F2 ) q(x) = 0 if x = 0; (F3 ) q(λx) ≤ q(x) for all x ∈ E and λ ∈ K with |λ| ≤ 1; (F4 ) q(x + y) ≤ q(x) + q(y) for all x, y ∈ E (subadditive); (F5 ) if λn → 0, then q(λn x) → 0 for all x ∈ E; (F6 ) if q(n ) → 0, then q(λxn ) → 0 for all λ ∈ K. (4) If q : E → R satisfies q(λx) = λq(x) for all λ ≥ 0, then q is called positively homogeneous. (5) If a seminorm (resp. F -seminorm) q satisfies q(x) = 0 implies x = 0, then q is called a norm (resp. F-norm) on E. Remarks. (1) If q is a seminorm or F -seminorm on a vector space E, it may happen that q(x) = 0 for some x 6= 0. Hence, if q is a seminorm or F -seminorm on a vector space E, then it defines a pseudometric dq on E given by dq (x, y) = p(x − y), x, y ∈ E. (2) It is easy to verify that if q is a seminorm on a vector space E, then, for each r > 0, the q-balls Bq (0, r) = {x ∈ E : q(x) < r}, Bq [0, r] = {x ∈ E : q(x) ≤ r} are absorbing, balanced and convex subset of E. Definition: Let A be an absorbing subset of a vector space E. Then the function ρA : E → R defined by ρA (x) = inf{r > 0, x ∈ rA}, x ∈ E, is called the Minkowski functional of A. Remark. It is easy to see that if, in addition, A is either balanced or convex, then (i) for any x ∈ E and ε > 0, x ∈ (ρA (x) + ε)A. (ii) {x ∈ E : ρA (x) < 1} ⊆ A ⊆ {x ∈ E : ρA (x) ≤ 1}. Lemma A.2.6. Let A be an absorbing subset of a vector space E. Then: (a) 0 ≤ ρA (x) < ∞ with ρA (0) = 0. (b) (i) ρA (tx) = tρA (x) for all x ∈ E and t ≥ 0 (positively homogeneous); (ii) A is balanced iff ρA (λx) = |λ|ρA (x) for all x ∈ E and λ ∈ K (absolutely homogeneous).


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(c) A is convex iff ρA (x + y) ≤ ρA (x) + ρA (y) for all x, y ∈ E

(subadditive).

Hence, if A is an absorbing, balanced and convex subset of a vector space E, then ρA is a seminorm on E. Note. If A is not convex, then ρA need not be subadditive. However, we have: Lemma A.2.7. Let A be an absorbing subset of a vector space E. If H is a balanced subset of E with H + H ⊆ A, then ρA (x + y) ≤ ρH (x) + ρH (y) for all x, y ∈ E. Proof. Let x, y ∈ E and let ε > 0. Then there exist r, s > 0 such that ε ρH (x) < r < ρH (x) + and x ∈ rH, 2 ε ρH (y) < s < ρH (y) + and y ∈ sH. 2 so s r H+ H] x + y ∈ rH + sH = (r + s)[ r+s r+s ⊆ (r + s)[H + H] (since H is balanced) ⊆ (r + s)A. Hence ρA (x + y) ≤ r + s ≤ ρH (x) +

ε ε + ρH (y) + = ρH (x) + ρH (y) + ε. 2 2

Since ε is arbitrary, ρA (x + y) ≤ ρH (x) + ρH (y). Lemma A.2.8. Let E be a vector space and p, q : E → R+ be positively homogeneous (e.g. p, q be seminorms on E or p = ρG , q = ρH be Minkowski functionals of neighborhoods G and H of 0 in a TVS E). Then p(x) ≤ q(x) ∀ x ∈ E ⇔ q(x) ≤ 1 implies p(x) ≤ 1 ∀ x ∈ E. Proof. (⇒) Suppose p(x) ≤ q(x) ∀ x ∈ E. If q(x) ≤ 1, then p(x) ≤ q(x) ≤ 1. (⇐) Suppose q(x) ≤ 1 implies p(x) ≤ 1 for all x ∈ E. Fix x ∈ E, we consider two cases:

252


x ) = q(x) = 1; and so, by Case I. Suppose q(x) 6= 0. Then q( q(x) q(x) hypothesis, p(x) x ) ≤ 1, or ≤ 1, or p(x) ≤ q(x). p( q(x) q(x)

Case II. Suppose q(x) = 0. Then q(x) < ε for every ε > 0, or x q( ) ≤ 1 for every ε > 0. ε Hence, by hypothesis, for any ε > 0, x p( ) < 1, or p(x) < ε ε Therefore, p(x) = 0. So, in this case p(x) = 0 = q(x). Thus, in each case, p(x) ≤ q(x) for all x ∈ E.

Lemma A.2.9. Let A be an absorbing and balanced (or convex) subset of a TVS (E, τ ). Then: ¯ (a) A0 ⊆ {x ∈ E : ρA (x) < 1} ⊆ A ⊆ {x ∈ E : ρA (x) ≤ 1} ⊆ A. (b) ρA is τ -continuous iff A is a τ neighborhood of 0. (c) If A is a τ neighborhood of 0, then: (i) A0 = {x ∈ E : ρA (x) < 1}. (ii) A¯ = {x ∈ E : ρA (x) ≤ 1}. We shall later see that if A is a “shrinkable” (not necessarily convex) τ neighborhood of 0, then ρA is τ -continuous on E. From the above lemmas, we obtain: Theorem A.2.10. If (E, τ ) is a locally convex space, then its topology τ can be defined by a family of seminorms, namely, the Minkowski functionals of balanced convex τ neighborhoods of 0. Conversely, if P is a family of seminorms on a vector space E over K, then P defines a locally convex topology τ (say) on E and a base of τ neighborhoods of 0 consists of all sets of the form {x ∈ E : max pj (x) ≤ ε}, , where ε > 0 and pj ∈ P, n ∈ N. 1≤j≤n

For general TVSs, we have: Theorem A.2.11. If (E, τ ) is TVS, then its topology τ can be defined by a family of F -seminorms. If τ is generated by a countable family of F -seminorms, then τ can be generated by a single F -norm and so it is metrizable. Proof. ([Wae71], p. 2-3) Let V ∈ W be any balanced τ neighborhood of 0 in E. Define V1 , ...., Vk , ... inductively in such a way that each Vk is balanced, V ⊇ V1 + V1 , and Vk ⊇ Vk+1 + Vk+1 . Let D = {p/2n : p is


rational, n ∈ N}, the set of dyadic rationals. Fix any t = P εk t < 1. Then t = N k=1 2N with εk = 0 or 1 for all k. Put Wt =

N X

εk Vk ; that is, Wt =

k=1

X

253 q 2N

∈ D, 0 ≤

Vk .

εk =1

Then Wt is balanced, and clearly Wt1 + Wt2 ⊆ Wt1 +t2 if t1 + t2 < 1. We put Wt = E for t ≥ 1, and define qV (x) = inf{t : x ∈ Wt }, x ∈ E. Then qV is an F -seminorm on E. Thus the family {qV : V ∈ W} generates the topology τ. If τ is generated by a countable family {qn : n ∈ N} of F -seminorms, then clearly τ can be generated by a single F -norm q given by ∞ X qn (x) , x ∈ E. q(x) = 2−n 1 + qn (x) n=1 Thus (E, τ ) is metrizable. The general criterion for the metrizability of a TVS may be stated as: Theorem A.2.12. (Metrization Theorem ) ([KN63], p. 48-49) (a) A TVS (E, τ ) is pseudo-metrizable iff it has a countable base of τ neighborhoods of 0. In this case, τ may be defined by a pseudo-metric d which is translation invariant and each ball about 0 is balanced; i.e., (i) d(x + z, y + z) = d(x, y) = d(x − y, 0) for all x, y, z ∈ E, (ii) d(λx, 0) ≤ d(x, 0) for all x ∈ E and λ ∈ K with |λ| ≤ 1; (b) A TVS (E, τ ) is metrizable iff it is Hausdorff and has a countable base of τ neighborhoods of 0. In this case, τ may be given by an F -norm q, where q(x) = d(x, 0) for all x ∈ E. Definition. A TVS E is said called locally bounded if E has a bounded neighborhood of 0. In this case, if V is a bounded neighborhood of 0 in E, then clearly, { n1 V : n ≥ 1} is a countable base of neighborhoods of 0 in E. Then, by above theorem, every Hausdorff locally bounded TVS is metrizable. Theorem A.2.13. (Kolmogoroff) (i) The topology of a Hausdorff locally bounded TVS E can always be generated by a k-norm for some k, 0 < k ≤ 1.

254


(ii) A Hausdorff TVS (E, τ ) is normable iff it has a bounded convex neighborhood of 0. (Hence every Hausdorff locally bounded locally convex space is normable.) Theorem A.2.14. Let E and F beTVSs. Then: (a) A linear mapping T : E → F is continuous on E iff T is continuous at 0 ∈ E iff T is uniformly continuous on E. (b) If f : E → K a non-zero linear functional, then f is continuous on E iff f −1 (0) is closed in E iff f is bounded on some neighborhood U of 0 in E. Definition: The set of all linear functionals on a vector space (or a TVS) E is a vector space, called the algebraic dual of E and is denoted by E ′ . If E is a TVS, the set of all continuous linear functionals on E is also a vector space, called the topological dual of E and is denoted by E ∗ . Clearly, E ′ is a vector space over K. Further, E ∗ is a vector subspace of E ′. Definition. Let (E, τ ) be a TVS and M a vector subspace of E. (1) The quotient space E/M is defined by E/M = {x + M : x ∈ E}, which is a vector space under the operations (x + M) + (y + M) = (x + y) + M,

λ(x + M) = λx + M,

where x, y ∈ E and λ ∈ K. (2) The canonical map (or quotient map) π : E → E/M is defined by π(x) = x + M, x ∈ E. (3) The quotient topology τM on E/M is defined by τM = {A ⊆ E/M : π −1 (A) ∈ τ }. Theorem A.2.15. ([Scha71], p. 20, 31) Let (E, τ ) be a TVS and M a vector subspace of E. Then: (a) (E/M, τM ) is a TVS. (b) π : (E, τ ) → (E/M, τM ) is a linear, continuous and open map. (c) τM is Hausdorff iff M is closed. (d) If (E, τ ) is locally convex whose topology is given by a family sE of seminorms, then (E/M, τM ) is also a locally convex space and its topology is given by the family {e p : p ∈ sE } of seminorms, where e p : E/M → R is given by e p(x + M) = inf{p(x + m) : m ∈ M}, x + M ∈ E/M.


255

(e) If (E, τ ) is metrizable (resp. normable) and M is closed, then (E/M, τM ) is also metrizable (resp. normable). (f) If (E, τ ) is complete and metrizable (resp. a Banach space) and M is closed, then (E/M, τM ) is complete and metrizable (resp. a Banach space). Remarks. (1) If M is a closed vector subspace of a TVS E, then the seminorm e p : E/M −→ R defined above in (d) becomes a norm on E/M, since e p(x + M) = 0 ⇒ x ∈ M = M ⇒ x + M = M, the zero of E/M.

(2) ([Rud91], p. 33) Let p be a seminorm on a vector space E and Np = ker (p) = p−1 (0). Then (i) inf{ p(x + m) : m ∈ Np } = p(x). (ii) If e p : E/Np −→ R is defined by e p(x + Np ) = p(x), x + Np ∈ E/Np , then e p is a norm on E/Np .

Definition. (1) If M be a proper maximal vector subspace of a vector space E and xo ∈ E, then the set H = xo + M is called a hyperplane through xo ; in particular, M is a hyperplane through 0. (2) Two vector subspaces M and N of E are said to be complementary if E = M ⊕ N (i.e. if E = M + N with M ∩ N = {0}). (3) The co-dimension of vector subspace M of E is defined as the dimension of its complementary subspace N (or equivalently, the dimension of the quotient space E/M). Theorem A.2.16. Let E be a vector space and H ⊆ E. Then: (a) H is a hyperplane iff H = ϕ−1 (α) = {x ∈ E : ϕ(x) = α} for some non-zero linear functional ϕ on E and α ∈ K. (b) H is a hyperplane through 0 iff M is of co-dimension one iff M = ϕ−1 (0) for some non-zero linear functional ϕ on E. (c) If E is a TVS, then a hyperplane H = ϕ−1 (α) in E is either closed or dense in E. Further, H = ϕ−1 (α) is closed iff ϕ is continuous on E. Theorem A.2.17. (Hahn-Banach) Let (E, τ ) be a TVS and M a vector subspace of E. (a) If f is a continuous linear functional on M and p a continuous seminorm on E with |f (x)| ≤ p(x) for all x ∈ M,

256


then there exists a g ∈ E ∗ such that g = f on M and |g(x)| ≤ p(x) for all x ∈ E. (b) If E is locally convex and f a continuous linear functional on M, then there exist a g ∈ E ∗ and a continuous seminorm p on E such that g = f on M and |g(x)| ≤ p(x) for all x ∈ E. Corollary A.2.18. Let E be a Hausdorff locally convex space (in particular, a normed space) over K. Then E ∗ separates points of E (i.e. for any xo 6= 0 in E, there exist a g ∈ E ∗ such that g(xo ) 6= 0). Note that if E is not Hausdorff or not locally convex, then E ∗ need not separate the points of E. Examples. For 0 < p < 1, the spaces ℓp (N) and Lp [a, b] are F spaces but not locally convex spaces (as seen before). However, (ℓp (N))∗ separates points of ℓp (N) and (Lp [a, b])∗ does not separate points of Lp [a, b] (as (Lp [a, b])∗ = {0}). Remark. In fact, if E is a TVS, then E ∗ 6= {0} iff E contains a non-empty open convex subset U 6= E ([Scha71], p. 47). For more material on non-locally convex spaces, see the monographs by Waelbroeck ([Wae71, Wae73]), Adasch, Ernst and Keim [AEK78], Khaleelulla [Khal81], Kalton, Peck and Roberts [KPR84], Rolewicz [Rol85]; see also the papers by Klee ([Kle60a, Kle60b]) and Kakol ([Kako85, Kako87, Kako90, Kako92]). Definition. Let (E, τ ) be a TVS. For any finite (or singleton) set A ⊆ E ∗ , the mapping pA : E → R defined by pA (x) = sup{|f (x)| : f ∈ A}, x ∈ E, is a seminorm pA on E. If A is the collection of all finite subsets of E ∗ , then the collection {pA : A ∈ A} of seminorms defines a locally convex topology on E, called the weak topology and denoted by w = w(E, E ∗ ) or σ(E, E ∗ ). Similarly, for any finite (or singleton) set B ⊆ E, the mapping qB : E ∗ → R defined by qB (f ) = sup{|f (x)| : x ∈ B}, f ∈ E ∗ . is a seminorm qB on E ∗ . If B is the collection of all finite subsets of E, then the collection {qA : B ∈ B} of seminorms defines a locally convex topology on E ∗ , called the weak ∗ topology on E and denoted by w ∗ = w(E ∗, E) or σ(E ∗ , E). Theorem A.2.19. ([Hor66], p. 185-209) (a) w(E, E ∗ ) ≤ τ.


257

(b) (E, w(E, E ∗)) is Hausdorff iff E ∗ separates points of E; in particular, if (E, τ ) is a Hausdorff locally convex space, then (E, w(E, E ∗ )) is Hausdorff. (c) (E, w(E, E ∗))∗ = (E, τ )∗ . (d) If (E, τ ) a Hausdorff locally convex space, then: (i) (Mazur) For any convex subset M of E, M is τ -closed iff M w τ is w(E, E ∗ )-closed; in particular, M = M . (ii) (Mackey) For any subset M of E, M is τ -bounded iff M is w(E, E ∗)-bounded. (e) If (E, τ ) is finite-dimensional, then τ = w(E, E ∗). We next consider strong and strong∗ topologies on E and E ∗ , respectively. We first note that, for any set B ⊆ E, the mapping qB : E ∗ → R defined by qB (f ) = sup{|f (x)| : x ∈ B}, f ∈ E ∗ , is a seminorm on E ∗ iff B is w(E ∗ , E)-bounded in E ∗ (see [RR64], p.45; [Hor66], p. 195). Definition. Let (E, τ ) be a TVS. For any w(E ∗ , E)-bounded set A ⊆ E ∗ , the mapping pA : E → R defined by pA (x) = sup{|f (x)| : f ∈ A}, x ∈ E, is a seminorm pA on E. If A is the collection of all w(E ∗, E)-bounded subsets of E ∗ , then the collection {pA : A ∈ A} of seminorms defines a locally convex topology on E, called the strong topology and denoted by β = β(E, E ∗). Clearly, w(E, E ∗) ≤ β(E, E ∗ ). Similarly, for any τ bounded (or equivalently, w(E ∗ , E)-bounded) set B ⊆ E, the mapping qB : E ∗ → R defined by qB (f ) = sup{|f (x)| : x ∈ B}, f ∈ E ∗ . is a seminorm qB on E ∗ . If B is the collection of all τ -bounded subsets of E, then the collection {qB : B ∈ B} of seminorms defines a locally convex topology on E ∗ , called the strong ∗ topology and denoted by β ∗ = β(E ∗ , E). Clearly, w(E ∗, E) ≤ β(E ∗ , E). Remarks. (1) ([Wila78], p. 119) If (E, || · ||) is a normed space, then the strong∗ topology β(E ∗ , E) on E ∗ is the norm topology given by ||f || = sup{|f (x)| : ||x|| ≤ 1, f ∈ E ∗ . However, in this case, the strong topology β(E, E ∗) on E need not the norm topology for E; in general, || · || ≤ β(E, E ∗ ). If E is a Banach space, then β(E, E ∗) = || · ||.

258


(2) The weak and strong topologies are often useful in giving counterexamples. For instance, the Banach space ℓ1 = ℓ1 (N) with the weak topology w = w(ℓ1, ℓ∞ ) is sequentially complete but not complete; ℓ1 with the weak∗ topology w ∗ = w(ℓ1, co ) is quasi-complete but not complete. (Recall that c∗o ∼ = ℓ1 and ℓ∗1 = ℓ∞ as normed spaces.) Hilbert space is weakly quasi-complete but not weakly complete. Definition: Let (E, τ ) be a TVS. A subset H of E ∗ is said to be equicontinuous if, given any ε > 0, there exists a neighborhood U of 0 in E such that |f (x)| < ε for all x ∈ U and f ∈ H. Definition. Let (E, τ ) is a TVS such that E ∗ separates points of E. For each set U ⊆ E, the set U p = {f ∈ E ∗ : qU (f ) = sup{|f (x)| : x ∈ U} ≤ 1} is called the polar of U in E ∗ . Theorem A.2.20. ([Hor66], p. 190, 195] Let (E, τ ) be a TVS and U ⊆ E. Then: ∗ (i) U p is convex, balanced and w(E ∗ , E)-closed in E . (ii) U p is absorbing iff U is w(E, E ∗ )-bounded iff qU is a seminorm on E ∗ . Theorem A.2.21. ([Hor66], p. 200]) Let (E, τ ) be a TVS. Then (i) [Hor66, p. 200] A subset H of E ∗ is equicontinuous iff there exists a τ -neighborhood U of 0 in E such that H ⊆ U p . In particular, for any τ neighborhood U of 0 in E, U p is an equicontinuous subset of E ∗ . (ii) Every equicontinuous subset H of E ∗ is w(E ∗ , E)-bounded. Theorem 1.2.22. (Bourbaki-Alaoglu) ([Hor66], p. 201) Let (E, τ ) be a TVS. Then any τ -equicontinuous subset H of E ∗ is relatively w(E ∗ , E)compact. In particular, for any τ neighborhood U of 0 in E, U p is w(E ∗, E)-compact. Corollary A.2.23. (Banach-Alaoglu) Let E be a normed space. Then the norm closed unit ball B ∗ [0, 1] = {f ∈ E ∗ : kf k ≤ 1} of E ∗ is w(E ∗ , E)-compact. Note. ([Ko69], p. 282) A norm closed unit ball of a normed space E ∗ need not be w(E, E )-compact in E; take, for example, E = ℓ1 .


259

Theorem A.2.24. ([Sim63], p. 234]) Let (E, ||·||) is a Banach space, and let S ∗ = (B1∗ , w ∗), a compact Hausdorff space. For any x ∈ X, define a continuous map x b : S ∗ → K by x b(f ) = f (x), f ∈ S ∗ .

Then the map π : x −→ x b is an isometric isomorphism of E onto a closed vector subspace of the Banach space C(S ∗ ).

Remarks. (1) By above theorem, any general Banach space (E, || · ||) can be considered as a closed vector subspace of the function space (C(X), || · ||∞ ), where X is a compact Hausdorff space. (2) [Jam74, p. 273] It is fair to ask whether C[0, 1] is typical of many other spaces C(X). In this regard, we mention a famous result of A.A. Milyutin (1966) which states that: if X is a uncountable compact metric space, then C(X) is isomorphic to C(I). Definition. A sequence {xn } in a TVS E is called a topological basis of E if, for each x ∈ E, there is a unique sequence {λn } of scalars such that k X λn xn (in the topology of E); x = lim k→∞

n=1

that is, for any neighborhood U of 0 in E, there exists an integer N ≥ 1 such that k X λi xi − x ∈ U for all k ≥ N. i=1

P In this case we write x = ∞ i=1 λi xi . For each m = 1, 2, ..., the linear functionals fm : E → K defined by fm (x) = λm for x =

∞ X

λi xi ∈ E,

i=1

are called the associated coefficient functionals of the basis{xn } . Note that fm (xn ) = δmn . If all fm are continuous, the basis {xn } is called a Schauder basis. Definition. (1) A TVS E is said to have the approximation property if the identity map on E can be approximated uniformly on precompact sets by continuous and linear maps of finite rank (i.e. with range contained in finite dimensional subspaces of E).

260


(2) A TVS E is said to be admissible if the identity map on E can be approximated uniformly on compact sets by continuous maps of finite rank. Every locally convex space and every F -space with a basis (e.g. ℓp , 0 < p < 1) is admissible ([Kl60a, Kl60b, Shu72]).

3. SPACES OF CONTINUOUS LINEAR MAPPINGS

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3. Spaces of Continuous Linear Mappings Definition. Let E and F be TVSs over the same field K. Then a linear mapping T : E → F is called bounded if it maps bounded sets of E into bounded sets of F. Lemma A.3.A. Every continuous linear mapping T : E → F is bounded. Proof. Let S be a bounded set in E, and let W be a neighborhood of 0 in F . By continuity of T, T −1 (W ) is a neighborhood of 0 in E. Since S is a bounded set in E, there exists r > 0 such that S ⊆ λT −1 (W ) for all |λ| ≥ r. Then T (S) ⊆ λW for all |λ| ≥ r. Hence T (S) is a bounded set in F . In general, a bounded linear map T : E → F need not be continuous. We shall see below that, under some additional hypothesis, every bounded linear mapping T : E → F is continuous. Theorem A.3.2. Let E be a metrizable TVS and F any TVS. Then any bounded linear map T : E → F is continuous (i.e. T is continuous iff T is bounded). Proof. First suppose that E is a metrizable TVS and F any TVS, and let T : E → F be a bounded linear map. In this case, there exists a countable base {U1 , U2 , ...} of neighborhood of 0 in E. Let V1 = U1 , V2 = U1 ∩ U2 , ...., Vn = U1 ∩ U2 ∩ .... ∩ Un , ..... Then {V1 , V2 , ....} is also a base of neighborhood of 0 with V1 ⊇ V2 ⊇ .... If T is not continuous, there is some neighborhood W of 0 in F such that T −1 (W ) is not a neighborhood of 0 in E. Hence, for each n ≥ 1, Vn " nT −1 (W ) and so Vn \nT −1 (W ) 6= ∅. For each n ≥ 1, choose xn ∈ Vn \ nT −1 (W ). Then {xn } is a bounded set in E. [Indeed, let V be any neighborhood of 0 in E. There exists a Vm with Vm ⊆ V . Now, for any n ≥ m, xn ∈ Vn ⊆ Vm . Choose r > 1 such that {x1 , ...., xm−1 } ⊆ rVm . Hence xn ∈ rVm ⊆ rV for all n ≥ 1.] However, {T xn } is not bounded in F since, for all n ≥ 1, xn ∈ / nT −1 (W ) and hence T (xn ) ∈ / nW . This contradicts the fact that T is bounded. Notations. The set of all bounded (resp. continuous) linear mappings T : E → F is denoted by BL(E, F ) (resp. CL(E, F )). Clearly, BL(E, F ) and CL(E, F ) are vector spaces with the usual pointwise operations and CL(E, F ) ⊆ BL(E, F ). Note that E ′ = L(E, K) and E ∗ = CL(E, K). Further, if F = E, BL(E) = BL(E, E) and CL(E) =

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CL(E, E) are algebras under composition (i.e. (ST )(x) = S(T (x))) and have the identity I : E → E given by I(x) = x (x ∈ E). Definition. (1) A TVS E is called ultrabornological [Iya68] if every bounded linear map from E into any TVS F is continuous. In this case, CL(E, F ) = BL(E, F ). By above theorem, every metrizable TVS is ultrabornological. (2) A TVS (E, τ ) is called ultrabarrelled ([RobW58, Edw65, Hus65]) if, any linear topology τ ′ on E, having a base of neighborhoods of 0 formed of τ -closed sets, is weaker than τ. Equivalently, a TVS E is ultrabarrelled if every closed linear map from E into any complete metrizable TVS is continuous [Iya68]. (3) E is called quasi-ultrabarrelled if every bounded linear map from E into any complete metrizable TVS is continuous. Every Baire TVS (in particular, F -space) is ultrabarrelled ([Edw65], p. 428). Further, every ultrabornological and every ultrabarrelled TVS is quasi-ultrabarrelled. We next consider some linear topologies on CL(E, F ), known as the G-topologies. Definition. Let E and F be TVSs, and let G be a collection of subsets of E and WF a base of closed balanced neighborhoods of 0 in F . We topologize CL(E, F ), as follows. For any D ∈ G and W ∈ WF , let U(D, W ) = {T ∈ CL(E, F ) : T (D) ⊆ W }. Then the family {U(D, W ) : D ∈ G, W ∈ WF } forms a subbase for a topology on CL(E, F ), called the topology of uniform convergence on members of G, or briefly, the G-topology and is denoted by tG . Note that, for any D ∈ G and W ∈ WF , (i) U(D, λW ) = λU(D, W ) (λ ∈ K); (ii) U(D, W ) is balanced (resp. convex) if W is so; (ii) U(D, W ) is absorbing iff T (D) is absorbed by W for each T ∈ CL(E, F ). Theorem A.3.4. ([Scha71], p. 79-80) (a) (CL(E, F ), tG ) is a TVS iff T (D) is bounded in F for each D ∈ G and T ∈ CL(E, F ). In particular, if G is the collection of all bounded subsets of E, then (CL(E, F ), tG ) is a TVS. (b) In addition, if E = ∪{D : D ∈ G}, then (CL(E, F ), tG ) is Hausdorff; if F is locally convex, so is (CL(E, F ), tG ). The most important cases of the G-topology are as follows:


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Definition. (1) If G is the collection of all bounded subsets of E, then the G-topology on CL(E, F ) is called the topology of uniform convergence on bounded sets ( or the uniform topology) and is denoted by tu . (2) If G is the collection of all precompact subsets of E, then the G-topology on CL(E, F ) is called the topology of precompact convergence and is denoted by tpc . (3) If G is the collection of all compact subsets of E, then the Gtopology on CL(E, F ) is called the topology of compact convergence and is denoted by tc . (4) If G is the collection of all finite subsets of E, then the G-topology on CL(E, F ) is called the topology of pointwise convergence and is denoted by tp . Clearly, tp ≤ tc ≤ tpc ≤ tu . Since each T ∈ CL(E, F ) maps bounded sets of E into bounded sets in F , it follows from the above theorem that CL(E, F ) with each of these topologies is a TVS. We also mention that, if F = K, we have CL(E, F ) = CL(E, K) = E ∗ and so the weak topology w(E ∗, E) and the strong topology β(E ∗ , E) on E ∗ are particular cases of the G-topologies. Theorem A.3.5. ([Edw65], p. 87; [Sch71], p. 85) Let E and F TVSs. (a) If E is ultrabornological (in particular, metrizable) and F is complete (resp. quasi-complete), then (CL(E, F ), tu ) is complete (resp. quasi-complete). (b) If E is a Baire space (in particular, an F-space) and F is quasicomplete, then (CL(E, F ), tu ) is quasi-complete. Proof. We only prove the first part of (a). Let {Tα : α ∈ I} be a Cauchy net in (CL(E, F ), tu ). Then, for each x ∈ E, {Tα (x) : α ∈ I} is a Cauchy net in F , as follows. [Let x ∈ E, and consider any W ∈ WF . Then U({x}, W ) is a tu -neighborhood of 0 in CL(E, F ). Since {Tα } is a Cauchy net in (CL(E, F ), tu ), there exists αo ∈ I such that Tα (x) − Tβ (x) ∈ W for all α, β ≥ αo . Hence {Tα (x)} is a Cauchy net in F .] Since F is complete, the mapping T : E → F given by T (x) = lim Tα (x), x ∈ E, α

t

u is well-defined. Clearly T is linear. To show that Tα −→ T , let D be a bounded set in E and W ∈ WF . Since {Tα } is tu -Cauchy, there exists αo ∈ I such that

Tα (x) − Tβ (x) ∈ W for all x ∈ D and α, β ≥ αo .

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Since W is closed, fixing α ≥ αo and taking lim, we have β

Tα (x) − T (x) ∈ W for all x ∈ D and α ≥ αo . Hence Tα − T ∈ U(D, W ) for all α ≥ αo . tu

Therefore Tα −→ T . We next show that T is continuous. Since E is ultrabornological, by definition, CL(E, F ) = BL(E, F ) and so it suffices to show that T is bounded (i.e. T maps bounded sets into bounded sets). Let D be any bounded set in E. To show that T (D) is bounded in F , let W ∈ WF . Choose a closed and balanced V ∈ WF such that V + V ⊆ W . Since tu Tα −→ T , there exists αo ∈ I such that Tα (x) − T (x) ∈ V for all x ∈ D and α ≥ αo . Since Tαo , being continuous, is bounded, there exists r ≥ 1 such that Tαo (D) ⊆ rV. Now, for any x ∈ D, T (x) = (T (x) − Tαo (x)) + Tαo (x) ∈ V + rV ⊆ r(V + V ) ⊆ rW ; that is, T (D) ⊆ rW . Hence T is bounded or, equivalently, T is continuous. Thus T ∈ CL(E, F ), and so (CL(E, F ), tu ) is complete. Remark. In general, (CL(E, F ), tp ) need not be complete. In fact, completeness results in the other G-topologies are mostly of negative character. If (E, τ ) is a locally convex space, then (E ∗ , w(E ∗ , E)) is complete iff E is finite dimensional; hence, if E is an infinite dimensional metrizable locally convex space, then (E ∗ , w(E ∗ , E)) is never complete. However, if E is complete, then (E ∗ , w(E ∗ , E)) is quasi-complete. Definition: Let E and F be TVSs. Then a subset F of CL(E, F ) is said to be (i) equicontinuous if for each neighborhood V of 0 in F , there exists a neighborhood U of 0 in E such that T (U) ⊆ V for all T ∈ F ; (ii) uniformly bounded if, each bounded set B in E, ∪{T (B) : T ∈ F } is a bounded set in F ; (iii) pointwise bounded if, each x ∈ E, {T (x) : T ∈ F } is a bounded set in F. Clearly, if F ⊆ CL(E, F ) is uniformly bounded, then it is pointwise bounded. Further, we obtain: Lemma A.3.6. ([Scha71], p. 83, 85) Let E and F be T V Ss, and let F ⊆ CL(E, F ) be equicontinuous. Then:


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(a) F is uniformly bounded (i.e. tu -bounded). (b) The topologies tpc and tp coincide on F . Proof. (a) Let D be a bounded set in E andW a neighborhood of 0 in F . Since F is equicontinuous, there exists a neighborhood G of 0 in E such that T (G) ⊆ W for all T ∈ F . Since D is a bounded set in E, there exists r > 0 such that D ⊆ rG. Then T (D) ⊆ rT (G) ⊆ rW for all T ∈ F . Thus ∪{T (D) : T ∈ F } is a bounded set in F . (b) Since tp ⊆ tpc , we need to prove that tpc ⊆ tp on F . Let To ∈ F , and let K be a precompact subset of E and W ∈ WF . We show that there exists a finite set D ⊆ K and V ∈ WF such that [To + U(D, V )] ∩ F ⊆ To + U(K, W ).

(1)

Choose a balanced V ∈ W with V +V +V ⊆ W. Since F is equicontinuous at 0, there exists a balanced neighborhood G of 0 in E such that T (G) ⊆ V for all T ∈ F .

(2)

Now K is precompact, there exists a finite set D = {yi : 1 ≤ i ≤ m} ⊆ K such that K ⊆ D + G. Now, let T ∈ [To +U(D, V )]∩F . Then T = To +S, where S ∈ U(D, V )]∩ F . By (2) and (3), S(K) ⊆ S(D + G) ⊆ S(D) + S(G) ⊆ V + (T − To )(G) ⊆ V + V + V ⊆ W. Thus T = To + S ∈ To + U(K, W ). This proves (1).

Theorem A.3.7. (Principle of uniform boundedness) [RobW58] Let (E, τ ) be an ultrabarrelled TVS and F any TVS. Let F ⊆ CL(E, F ) be a collection such that, for each x ∈ E, the set F (x) = {T (x) : T ∈ F } is bounded in F . Then F is equicontinuous; hence, for any bounded set D in E, ∪{T (D) : T ∈ F } is a bounded set in F . Proof. ([Edw65], p. 465) Let WE and WF denote a base of closed neighborhoods of 0 in E and F , respectively. (Let V be a closed neighborhood of 0 in F . We show that there exists a neighborhood V ∗ of 0 in E such that T (V ∗ ) ⊆ V for all T ∈ F .) For each V ∈ WF , let V ∗ = ∩{T −1 (V ) : T ∈ F }.

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Then V ∗ is τ -closed and absorbing in E, as follows. [Since the inverse image of a closed set under a continuous mapping is a closed set and also the intersection of any number of closed sets is closed, it follows that V ∗ is τ -closed in E. Since F is pointwise bounded, for each x ∈ E, there exists rx ≥ 1 such that T (x) ∈ rx V for all T ∈ F .

(2)

i.e. x ∈ rx (∩{T −1 (V ) : T ∈ F }) = rx V ∗ . This implies that V ∗ is absorbing.] Then the collection U = {V ∗ : V ∈ WF } may be taken as a base of neighborhoods of 0 for a vector topology τ ∗ on E. Since (E, τ ) is ultrabarrelled, τ ∗ ≤ τ and so each V ∗ is a neighborhood of 0 in E. Thus T (V ∗ ) ⊆ V for all T ∈ F , and so F is equicontinuous. Theorem A.3.8. (Banach-Steinhaus) Let E be an ultrabarrelled TVS and F is a quasi-complete TVS. Let {Tα : α ∈ I} be a net in CL(E, F ) such that, for each x ∈ E, T (x) = lim Tα (x) α

exists. Then T is continuous on E and Tα → T uniformly on precompact subsets of E. Further, if the net {Tα : α ∈ I} is replaced by a sequence {Tn }, it is enough that F be sequentially complete. Proof. (see [Edw65], p. 465) It is easy to verify that T is linear. Since {Tα : α ∈ J} is pointwise bounded, by the principle of uniform boundedness, {Tα : α ∈ J} is equicontinuous. We now show that T is continuous. Let W be a closed neighborhood of 0 in F . Since {Tα : α ∈ J} is equicontinuous, there exists a neighborhood U of 0 in E such that Tα (U) ⊆ W for all α ∈ J. Since W is closed, for each x ∈ U, limα Tα x ∈ W ; that is, for each x ∈ U, T x ∈ W . So T (U) ⊆ W and hence T is continuous. Next, by Lemma A.3.6(b), the topologies tpc and tp coincide on A; hence Tα → T uniformly on precompact subsets of E. We next state the open mapping and the closed graph theorems. Definition. Let E and F be TVSs. A mapping T : E → F is called open if T (U) is open in F for every open set U in E. T is called open at x ∈ E if T (U) contains a neighborhood of T (x) whenever U is a neighborhood of x in E. It is clear that T is open iff T is open at every point of E. Because of the invariance of vector topologies, it follows that a linear mapping T : E → F is open iff it is open at 0 ∈ E.


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Theorem A.3.9. (Open mapping theorem) [RobW58] Let E be an F-space and F an ultrabarrelled TVS, and let T : E → F be a continuous and onto linear mapping. Then T is an open mapping. If, in addition T is one-one, then T −1 : F → E is also continuous. Proof. See ([Edw65], p. 437; [Hus65, p. 53]). As before, a mapping T : E → F is said to have a closed graph if its graph G(T ) = {(x, T x) : x ∈ E} is closed in E × F or, equivalently, for any net {xα } in E with xα → x ∈ E and T (xα ) → y ∈ F, we have y = T (x). Every continuous map T : E → F has a closed graph. The converse holds under some additional hypotheses in the following form. Theorem A.3.10. (Closed graph theorem) [RobW58] Let E be an ultrabarrelled TVS and F an F-space. If T : E → F is linear and has closed graph, then T is continuous. Proof. See ([Edw65], p. 437; [Hus65], p. 53).

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4. Shrinkable Neighborhoods in a TVS In this section, we study the notion of a shrinkable neighborhood of 0 in a TVS E, as given by Klee [Kl60a, Kl60b]. Definition. A neighborhood W of 0 in a TVS E is said to be shrinkable if tW ⊆ W 0 for all 0 ≤ t < 1. Lemma A.4.1. Any convex neighborhood V of 0 in a TVS E is shrinkable. Proof. Let ρV be the Minkowski functional of V . Then, since ρV is continuous, we have V 0 = {x ∈ E : ρV (x) < 1}, V = {x ∈ E : ρV (x) ≤ 1}. Let y ∈ V and 0 ≤ t < 1. Now 1 − ρV (ty) ≥ ρ(y) − tρV (y) = (1 − t)ρV (y) > 0. Hence ρV (ty) < 1 and so ty ∈ V 0 . Thus tV ⊆ V 0 . Important facts about shrinkable neighborhoods are: (A) Every Hausdorff TVS E has a base of shrinkable neighborhoods of 0. (B) If W is a shrinkable neighborhood of 0 in a TVS E, then its Minkowski functional ρW is continuous and positively homogeneous. The proofs of these important results, due to Klee [Kl60a], may not be found in the text books and therefore we include them here. We first need to prove the following three results. Theorem A.4.2. Every Hausdorff TVS is linearly homeomorphic with a linear subspace of a product of metric linear spaces. Proof. Let E be a Hausdorff TVS and W the class of all open balanced neighborhoods W of 0 in E. For each W ∈ W, let Wo , W1 , W2 , .... be a sequence of members of W such that Wo = W, and Wn + Wn ⊆ Wn−1 for n = 1, 2, .... ∞

Let MW = ∩ Wn . Then clearly n=o

MW = [−1, 1]MW and MW + MW ⊆ MW , and so MW is a linear subspace of E. Let πW be the quotient map of E onto E/MW . Let E/MW be topologized by taking {πW (Wn ) : n ≥ 1} as a base of neighborhoods of 0. It is easily verified that E/MW is a TVS and πW is continuous. Further, −1 πW (πW (Wn )) = Wn + MW ⊆ Wn + Wn ⊆ Wn−1 ,

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∞

and so it follows that ∩ πW (Wn ) = {MW }. Thus the topology of E/MW n=o is Hausdorff; further it is metrizable since it has a countable base of neighborhoods of 0. Let P = Π E/MW (the product space), let π : E → P be defined by

W ∈W

(π(x))W = πW (x), x ∈ E, W ∈ W. Then π(E) is a linear subspace of P and π is a linear homeomorphism. This next two lemmas are easy to verify. Lemma A.4.3. Suppose E is a Hausdorff TVS, M a vector subspace of E, and W a shrinkable neighborhood of 0 in E. Then W ∩ M is a shrinkable neighborhood of 0 in M. Lemma A.4.4. Suppose {Eα : α ∈ I} is a family of Hausdorff TVSs, Wα is a shrinkable neighborhood of 0 in Eα for each α ∈ I, and Wα = Eα for all but finitely many α ∈ I. Then the set Π Wα is a a∈I

shrinkable neighborhood of 0 in the product space Π Eα . a∈I

Theorem A.4.5. In a Hausdorff TVS E, every neighborhood of 0 contains a shrinkable neighborhood of 0. Proof. In view of the above results, we may assume that E is a metrizable TVS. According to a classical result of Edelheit and Mazur (see [DS58], p.91), the topology of E can be generated by an invariant metric d which is strictly monotonic in the sense that d(tx, 0) < d(sx, 0) for 0 ≤ t < s and x 6= 0. We need to show that, for any ε > 0, B(0, ε) = {x ∈ E : d(x, 0) < ε} is shrinkable. [Let y ∈ B(0, ε) and 0 ≤ t < 1. To show that ty ∈ [B(0, ε)]0, we show that there exists a δ > 0 such that B(ty, δ) ⊆ B(0, ε). Since t < 1, d(ty, 0) < d(y, 0) and so δ = d(y, 0)−d(ty, 0) > 0. Let z ∈ B(ty, δ). Then d(z, 0) ≤ d(z, ty) + d(ty, 0) < δ + d(ty, 0) = d(y, 0) < ε. Hence z ∈ B(ty, δ) ⊆ B(0, ε). Thus ty ∈ [B(0, ε)]0 and so B(0, ε) is shrinkable.] Theorem A.4.6. Let W be a balanced neighborhood of 0 in a Hausdorff TVS E and let ρ = ρW be the Minkowski functional of W . Then: (a) ρ is upper semicontinuous iff W 0 = {x ∈ E : ρ(x) < 1}; (b) ρ is lower semicontinuous iff W = {x ∈ E : ρ(x) ≤ 1}. (c) ρ is continuous iff W is shrinkable.

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Proof. We first note that W 0 ⊆ {x ∈ E : ρ(x) < 1} ⊆ W ⊆ {x ∈ E : ρ(x) ≤ 1} ⊆ W .

(1)

(a) From (1) it follows that W 0 = {x ∈ E : ρ(x) < 1} iff the latter set is open. Since ρ is positively homogeneous, openness of the set {x ∈ E : ρ(x) < 1} is equivalent to that of the set {x ∈ E : ρ(x) < t} for all t > 0, and that in turn is equivalent to the upper semicontinuity of ρ. (b) Again by (1) we note that W = {x ∈ E : ρ(x) ≤ 1} iff the latter set is closed. Since ρ is positively homogeneous, closedness of the set {x ∈ E : ρ(x) ≤ 1} is equivalent to that of the set {x ∈ E : ρ(x) ≤ r} for all r > 0, and that is equivalent to the lower semicontinuity of ρ. (c) Suppose ρ is continuous. To show that W is shrinkable, let 0 ≤ t < 1. Then y

∈ ⇒ ⇒ ⇒

t.W = t{x : ρ(x) ≤ 1} y = tx, ρ(x) ≤ 1 ρ(y) = tρ(x) < 1 y ∈ {z ∈ E : ρ(x) < 1} = W 0 ,

as required. Conversely, suppose that W is shrinkable. We first show that ρ is lower semicontinuous or that A = {x ∈ E : ρ(x) < 1} is open. Let y ∈ A. Let t = max{ 21 , ρ(y)} < 1. Then 1 1 ρ( y) = ρ(y) ≤ 1, and so y ∈ tW ⊆ W 0 ⊆ A. t t Hence A is open. To establish upper semicontinuity of ρ, we need to show that {x ∈ E : ρ(x) > 1} is open or that B = {x ∈ E : ρ(x) ≤ 1} is closed. We claim that B = W . Clearly, B ⊆ W . Suppose W B, and 1 1/2 let z ∈ W \B. Then r = ρ(z) > 1, and so ρ( r1/2 z) = (r) > 1. Now, 1 since r1/2 < 1 and W is shrinkable, 1 r 1/2

z∈

1 r 1/2

W ⊆ W 0 ⊆ {x ∈ E : ρ(x) < 1},

1 and so ρ( r1/2 z) < 1, a contradiction. Hence B = W , and so B is closed. Thus ρ is both lower and upper semicontinuous, and hence it is continuous.

Theorem A.4.7. Let W be a closed shrinkable neighborhood of 0 in a TVS E, ρ = ρW the Minkowski functional of W , and h : E → E is given by x if x ∈ W, h(x) = x if x ∈ E\W. ρ(x)

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271

Then h is continuous. Proof. Let A = W = {x ∈ E : ρ(x) ≤ 1} and B = {x ∈ E : ρ(x) ≥ 1}. Then A and B are closed with E = A ∪ B. Define f : A → R and g : B → R by f (x) = x, x ∈ A, x g(x) = , x ∈ B. ρ(x) Hence, by the Map Gluing Theorem, h is continuous on E.

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5. Non-Archimedean Functional Analysis In this section, we consider the notions of non-Archimedean TVSs, locally F-convex spaces and almost non-Archimedean TVSs. Definition. Let F be any field with unit element e. Then a function | · |: F → R is called a valuation on F if, for any a, b ∈ F, (i) |a| ≥ 0, with |a| = 0 iff a = 0; (ii) |ab| = |a|.|b|; (iii) |a + b| ≤ |a| + |b|. In this case, the pair (F, | · |) is called a valued field. Clearly, |e| = 1. Definition. A valued field (F, | · |) is called a non-Archimedean field (in short, an NA field ) if |a + b| ≤ max{|a|, |b|} for all a, b ∈ F (Strong Triangular inequality). (ST) Remark. Recall that an important property of the field R is that it has the Archimedean Property in the sense that, given any x > 0 in R, there exists an n ∈ N such that x ≤ n; equivalently, N is not bounded in R. The proof of this fact is based on the supremum (or completeness) axiom of R. However, if (F, | · |) is an NA valued field, then, using (ST) inequality, for any integers n ∈ N, | ne |=| e + e + ....... + e |≤ max{| e |, | e |, ..., | e |} =| e |= 1; hence the set Ne is bounded in (F, | · |). Thus (F, | · |) does not have the Archimedean property. This justifies the use of the term nonArchimedean for such fields. The formula (ST) is essential to the entire non-Archimedean theory. Examples (1) Clearly, R and C with the usual absolute value are valued fields but not non-Archimedean. (2) Any field F with the trivial valuation 1 if a 6= 0 | a |= 0 if a = 0

is an NA field. All other valuations on F are called non-trivial valuations. (3) Let F = Q, the field of rationals, and let p be any fixed prime number. Every x ∈ Q, x 6= 0,can be written uniquely as x = pk · ab , where k ∈ Z = {0, ±1, ±2, ........} and a, b cannot be divided by p. Define x →| x |p by −k p if x 6= 0, | x |p = 0 if x = 0.

5. NON-ARCHIMEDEAN FUNCTIONAL ANALYSIS

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Then | . |p is an NA valuation on Q, called the p-adic valuation on Q. Note that (Q, | x |p ), being a metric space, has a completion, denoted by Qp . Then Qp is called the field of p-adic numbers. Remark. Let E be a vector space over a valued field (F, | · |). Then the definition of a TVS E over F is identical with that of a TVS E over K. However, the result on the characterization of a base of neighborhoods of 0 with the desired properties needs some modification. In fact, in most cases, we require that the valued field (F, | . |) is non-trivial. The reason is that, for any real r > 0, we can choose a, b ∈ F such that 0 < |a| ≤ r and |b| ≥ r. [Choose c ∈ F such that |c| = 6 0 and |c| = 6 1. Clearly, if −1 −1 0 < |c| ≤ r, then |c | ≥ r; if |c| ≥ r, then 0 < |c | ≤ r.] Definition. A subset S of vector space E over F is called a nonArchimedean set (in short, an NA set) if S + S ⊆ S. By induction, for any k ∈ N, the k-fold sum S + S + ......... + S ⊆ S. A TVS (E, τ ) over a valued field (F, | · |) is called a non-Archimedean TVS (in short, an NA TVS) if it has a base of τ neighborhoods of 0 consisting of NA sets. Examples. (1). Every NA valued field (F, | · |) is an NA TVS over itself. (2). Any indiscrete TVS is an NA TVS. (3). The trivial TVS E = {0} is an NA TVS. Theorem A.5.3. Let (E, τ ) be a TVS over a valued field (F, | · |) . If V is any NA neighborhood of 0 in E, then V is clopen (i.e., both closed and open). Proof. V is open, as follows. Let y ∈ V . Then y + V ⊆ V + V ⊆ V (since V is NA), so y has a neighborhood y + V with y + V ⊆ V. Hence V is open. V is closed, as follows. We need to show that E \ V is open. Let x ∈ E\V. Then V ∩ (x − V ) = ∅. [Suppose V ∩ (x − V ) 6= ∅, and let z ∈ V ∩(x − V ) . Then z = y1 , z = x−y2 with y1 , y2 ∈ V, and so y1 = x−y2 . Hence x = y1 + y2 ∈ V + V ⊆ V, a contradiction since x ∈ E \ V .] Therefore, x − V ⊆ E\V. Since x − V is a neighborhood of x, E \ V is open. Thus V is closed. Recall that a topological space X (6= ∅) is said to have dimension 0 at a point x ∈ X if x has a local base Bx of neighborhood consisting of

274


sets which are both open and closed; X is said to be 0-dimensional if X has a dimension 0 at each point of X. Theorem A.5.4. Every NA TVS (E, τ ) is a 0-dimensional topological space. Proof. Since (E, τ ) is an NA TVS, it has a base W of τ neighborhoods of 0 consisting of NA sets. Then, by above theorem, each V ∈ W is both open and closed. Hence, for each x ∈ E, Bx = {x + V : V ∈ W} consists of both open and closed neighborhoods of x. Theorem A.5.5. Let (E, τ ) be an NA TVS over a non-trivially valued field (F, | · |). Then there exist a base of neighborhoods of 0 consisting of NA (hence clopen) ’balanced’ sets.

6. TOPOLOGICAL ALGEBRAS AND MODULES

275

6. Topological Algebras and Modules Definition. Let A be an algebra over K (=R or C) and τ a topology on A. Then the (algebra) multiplication (x, y) → xy from A × A → A is called: (i) separately continuous if, for any τ neighborhood W of 0 in A and any x ∈ A, there exists a τ neighborhood V of 0 in A such that xV ⊆ W and V x ⊆ W . (ii) hypocontinuous if, for any τ neighborhood W of 0 in A and any τ -bounded set D in A, there exists a τ neighborhood V of 0 in A such that DV ⊆ W and V D ⊆ W . (iii) jointly continuous if, for any τ neighborhood W of 0 in A, there exists a τ neighborhood U of 0 in A such that U 2 ⊆ W, where U 2 = U.U = {xy : x, y ∈ U}. Clearly, joint continuity ⇒ hypocontinuity ⇒ separate continuity. Definition: Let A be an algebra over K (=R or C) and τ a topology on A such that (A, τ ) is a TVS. Then the pair (A, τ ) is called a topological algebra if it has a separately continuous multiplication. A complete metrizable topological algebra is called an F -algebra. Recall that a topological space X is called a Baire space if whenever X = ∪∞ n=1 An with each An closed, then, for at least one m, int(Am ) 6= ∅. Theorem A.6.1. (Arens [Are47]) Every Baire metrizable topological algebra (A, τ ) has jointly continuous multiplication. In particular, every F -algebra (A, τ ) has jointly continuous multiplication. Proof. ([Hus83], p. 5-6) Since (A, τ ) is metrizable, there exists a metric d on A such that τ = τd . To show that the map (x, y) −→ xy is jointly continuous at (0, 0) ∈ A × A, let W be any τd -neighborhood of 0 in A. We shall find a τd -neighborhood U of 0 in A such that U.U ⊆ W . For each n ≥ 1, let B(0, n1 ) = {x ∈ A : d(x, 0) < n1 }. Then {B(0, n1 )} is a countable base of τd -neighborhoods of 0. Choose a closed balanced τ neighborhood V of 0 in A such that V + V ⊆ W . Let An = {x ∈ A : xB(0, n1 ) ⊆ V }. Clearly An B(0, n1 ) ⊆ V . We now show that (i) each An is closed; (ii) A = ∪∞ n=1 An . (i) An is closed. [Let x ∈ An . Then there exists{xk } ⊆ An such that τd xk −→ x. Then xk B(0, n1 ) ⊆ V for all k ≥ 1. By separate continuity of multiplication, for any y ∈ B(0, n1 ), xk y −→ xy ∈ V . SinceV is closed, V = V , and so xy ∈ V . Therefore xB(0, n1 ) ⊆ V , and so x ∈ An . Hence An is closed.]

276


∞ ∞ (ii) A = ∪∞ n=1 An . [Clearly ∪n=1 An ⊆ A. To show A ⊆ ∪n=1 An , let x ∈ A. Since {B(0, n1 )} is a base of τd -neighborhoods of 0, by separate continuity of (x, y) −→ xy at (x, 0), there exists a B(0, N1 ) ∈ {B(0, n1 )}such that xB(0, N1 ) ⊆ V . Hence x ∈ AN ⊆ ∪∞ n=1 An . This ∞ proves that A ⊆ ∪n=1 An .] Since A is a Baire space and A = ∪∞ n=1 An with each An closed, by the Baire Category theorem, int(Am ) 6= ∅ for some m ≥ 1. Let xo ∈ int(Am ). Then there exists ε > 0 such that

B(xo , ε) ⊆ Am .

(1)

Now U = B(0, ε) ∩ B(0, m1 ) is a τd -neighborhood of 0 in A. We now show that U.U ⊆ W . Clearly, U.U ⊆ B(0, ε)B(0, m1 ). We need to show that B(0, ε)B(0, m1 ) ⊆ W. [For any x ∈ B(0, ε), d(x + xo , xo ) = d(x, 0) < ε and so, by (1), x + xo ∈ B(xo , ε) ⊆ Am ; hence, by definition of Am , xB(0,

1 1 1 1 ) = [(x + xo ) − xo ]B(0, ) ⊆ (x + xo )B(0, ) − xo B(0, ) m m m m ⊆ V − V ⊆ W.

So B(0, ε)B(0, m1 ) ⊆ W.] Thus U.U ⊆ W.

Note: (1) In general, separate continuity of multiplication need not imply joint continuity even in algebras which are normed spaces but not complete (see [Rud91], p. 247, 272). (2) Every ultrabarrelled topological algebra has hypocontinuous multiplication (see Theorem A.6.28 below). (3) [Fra05, p. 8] If (A, τ ) is a topological algebra, then its completion b A (i.e. the completion of the underlying TVS (A, τ )) need not be topological algebra unless the multiplication of (A, τ ) is jointly continuous. (4) In some books/papers, a topological algebra is assumed to have jointly continuous multiplication; but we shall not use this notion here. Definition. A subset U of an algebra A is called idempotent (or multiplicative or an m-set) if U 2 ⊆ U. Definition. A topological algebra (A, τ ) is called: (i) a locally convex algebra if it has a base W=WA of neighborhoods of 0 consisting of convex sets; or equivalently, its topology τ is generated


277

by a family sA of seminorms such that, for each p ∈ sA , there exists a p1 ∈ sA such that p(xy) ≤ p1 (x)p1 (y) for all x, y ∈ A; (ii) a locally m-convex algebra if it has a base W of neighborhoods of 0 consisting of convex and idempotent sets (i.e. the sets U such that U 2 ⊆ U); or equivalently, if its topology τ is generated by a family sA of seminorms which are submultiplicative, i.e. for each p ∈ sA , p(xy) ≤ p(x)p(y) for all x, y ∈ A; (iii) a locally idempotent algebra if it has a base W of neighborhoods of 0 consisting of idempotent sets; (iv) a locally bounded algebra if it has a bounded neighborhood of 0. Examples. (1) Every normed algebra and, more generally, every locally m-convex algebra is locally convex. (2) Every locally m-convex algebra is locally idempotent and every locally idempotent algebra is a topological algebra with jointly continuous multiplication. (3) The notion of locally idempotent algebras is a strict generalization of the notion of locally m-convex algebras. This is shown by the algebra ℓq+ (0 ≤ q < 1) consisting of all two-sided complex sequences a = {an } P p such that kakp = ∞ |a | is convergent for each p with q < p ≤ 1 (see n=1 n [Zel60], p.348). Theorem A.6.2. ( [Zel73], p. 32) Let (A, d) be an F -algebra with identity e. Then the following conditions are equivalent (a) There exists an equivalent metric ρ on A such that ρ(xy, 0) ≤ ρ(x, 0).ρ(y, 0) for all x, y ∈ A. (b) A is a locally bounded algebra. (c) The topology of A can be generated by a submultiplicative k-norm k.kk , 0 < k ≤ 1, with kekk = 1. Note. (1) If A is a locally m-convex algebra with identity e, then its topology can be given by a family {pα : α ∈ I} of submultiplicative seminorms such that pα (e) = 1 for all α ∈ I ([Fra05]. p. 16). (2) If A is a locally convex algebra with identity e, we do not know whether its topology can be given by a family {pα : α ∈ I} of seminorms which satisfy pα (e) = 1 for all α ∈ I. Definition: Let A be an algebra. A map ∗ : x → x∗ from A into A is called an involution on A if it satisfies:

278


(i) (x + y)∗ = x∗ + y ∗ , ∗ (ii) x∗ = x, (iii) (λx)∗ = λx∗ , λ is the complex conjugate of λ ∈ K, (iv) (xy)∗ = y ∗x∗ , (v) e∗ = e if A has identity e. An algebra A equipped with an involution ∗ is called an involutive algebra or a ∗-algebra. Definition. A normed algebra (A, ||.||) is called: (1) a normed ∗-algebra if A has an involution such that ||x∗ || = ||x|| for all x ∈ A; in this case, clearly the involution map x → x∗ is continuous. (2) a B ∗ -algebra (or equivalently, a C ∗ -algebra) if A is a Banach ∗algebra whose norm ||.|| satisfies ||xx∗ || = ||x∗ x|| = ||x||2 for all x ∈ A; In this case, the norm ||.|| is called a C ∗ -norm. Definition. A complete locally m-convex ∗-algebra (A, τ ) is called a locally C ∗ -algebra if its topology τ is determined by a family { pα : α ∈ I} of submultiplicative seminorms satisfying pα (x∗ x) = [pα (x)]2 for all x ∈ A. The seminorms pα are called C ∗ -seminorms. Example. (Arens Algebra) ([Are46]; [Mal86], p. 12) Let

∞

Lω [0, 1] = ∩ Lp [0, 1], p=1

where each Lp [0, 1], p ≥ 1, is a normed space with respect to the norm: ||f ||p = (o |f (t)|p dt)

1/p

, f ∈ Lp [0, 1].

Then: (a) Lω [0, 1] is an algebra and ||f ||1 ≤ ||f ||2 ≤ ...∀f ∈ Lω [0, 1]. Note that: ||f g||p ≤ ||f ||q ||g||r for 1/p = 1/q + 1/r; hence Lω [0, 1] is nonnormed algebra. (b) For any p ≥ 1 and ε > 0, let Bp (0, ε) = {f ∈ Lω [0, 1] : kf kp < ε}. Then the collection {Bp (0, ε) : p ∈ N, ε > 0} forms a base of neighborhoods of 0 for a Hausdorff locally convex topology τω on Lω [0, 1].


279

(c) (Lω [0, 1], τω ) is a complete metric space w.r.t. the metric: dw (f, g) =

∞ X p=1

2−p

||f − g||p , f, g ∈ Lω [0, 1]. 1 + ||f − g||p

(d) (Lω [0, 1], τω ) is not locally m-convex. Definition: Let A be an algebra with identity e and let x ∈ A. Then x is said to be invertible (or regular) if there exists some y ∈ A, called the inverse of x, such that xy = yx = e; in this case, we write y = x−1 . Definition: Let A be an algebra. For any x, y ∈ A, the q-product x ◦ y of x, y is defined by x ◦ y = x + y − xy. An element x ∈ A is said to be q-invertible if there exists some y ∈ A, called the q-inverse of x such that x ◦ y = y ◦ x = 0; in this case, we write y = x−1 . Definition: Let A be an algebra, and let x ∈ A. (1) If A has an identity e, then the spectrum of x is defined by σA (x) = {λ ∈ C : λe − x is not invertible in A}. (2) If A is without identity, then the spectrum of x is defined as x σA (x) = {λ ∈ C : λ 6= 0, is not q-invertible in A} ∪ {0}. λ (3) The spectral radius of x is defined as rA (x) = sup{|λ| : λ ∈ σA (x)}. Clearly, 0 ≤ rA (x) ≤ ∞, rA (0) = 0, rA (αx) = |α|rA (x) (α ∈ C), and rA (e) = 1 if A has an identity e. Example. Let X be a locally compact Hausdorff space. (1) If X is compact, for any f ∈ Co (X) = C(X), σC(X) (f ) = {f (x) : x ∈ X} = f (X). (2) If X is non-compact, then σCo (X) (f ) = f (X) ∪ {0} for all f ∈ Co (X); σC(X) (f ) = f (X) ⊆ f (βX) = f (X) = σCb (X) (f ) for all f ∈ Cb (X), where βX is the Stone-Cech compactification of X.

280


Definition. A topological algebra A with identity (resp. without identity) is said to be a Q-algebra if the set G(A) of all invertible elements of A (resp. Gq (A) of all quasi-invertible elements of A) is open in A. Examples. (1) Every complete locally bounded Hausdorff topological algebra (hence every Banach algebra) is a Q-algebra ([Mal86], p.42). (2). An incomplete normed algebra need not be a Q-algebra, since neither G(A) nor Gq (A) need be an open set. (3). A complete metrizable locally m-convex algebra need not be a Qalgebra (e.g.(C(R), k), with k the compact-open topology, is a complete metrizable locally m-convex algebra but not a Q-algebra ([Hus83], p. 8). Theorem A.6.3. ([Mal86], p. 43-44, 59) Let A be a topological algebra. Then the following are equivalent: (a) A is a Q-algebra, i.e. Gq (A) is open in A. (b) [Gq (A)]0 6= ∅. (c) Gq (A) is a neighborhood of 0 in A. (d) S(A) = {x ∈ A : rA (x) ≤ 1} is a neighborhood of 0 in A. While considering the spectrum and related notions in the sequel, we shall consider all algebras over the complex field C. Definition: A topological algebra A with identity (resp. without identity) is said to have a continuous inversion (resp. continuous quasiinversion) if the inversion map x → x−1 from G(A) → G(A) (resp. the quasi-inversion map x → x−1 from Gq (A) → Gq (A)) is continuous. In the sequel, for convenience, we shall consider mostly the results for the more general notion of ”quasi-inversion”. Theorem A.6.4. (a) ([Mal86], p. 52) Every locally m-convex algebra (in particular, every normed algebra) A has a continuous quasi-inversion. Conversely: (b) ([Pa78], p. 277) Let A be a commutative locally convex F -algebra with a continuous quasi-inversion. Then A is a locally m-convex algebra. Example. [Are46] The Arens algebra Lω [0, 1] = ∩ Lp [0, 1] is a lop≥1

cally convex F -algebra, but not a locally m-convex algebra and it does not have a continuous inversion. Definition: ([Mal86], p. 85; [Hus83], p.10) Let (A, τ ) be a locally m-convex algebra whose topology τ is generated by a family sA = {p : p ∈ sA } of submultiplicative seminorms. For each seminorm p ∈ sA , Np = p−1 (0) is a two-sided ideal of A and so A/Np is a normed algebra with the norm: ||πp (x)||p = p(x), where πp : A → A/Np is the canonical


281

map given by πp (x) = xp = x + Np , x ∈ A. \p denote the completion of the normed algebra A/Np . Let Ap = A/N Then Ap is a Banach algebra and each πp can be regarded as the map from A to Ap . Each πp is a continuous homomorphism. We define the partial ordering ≤ on sA by p, q ∈ sA , p ≤ q ⇔ pp (x) ≤ pq (x) for all x ∈ A. Then, for p ≤ q, we have Nq ⊆ Np and so there exists an algebra homomorphism ϕpq : A/Nq → A/Np which has the unique extension to ϕ epq : Aq → Ap for p ≤ q, where ϕ epq (πq (x)) = πp (x) for all x ∈ A and p ≤ q. Clearly, each ϕpq is a continuous homomorphism. Let

lim A/Np = {(πp (x)) ∈ Πp∈sA A/Np : ϕpq (πq (x)) = πp (x) for p ≤ q, x ∈ A}, ←−

lim Ap = {(πp (x)) ∈ Πp∈sA Ap : ϕ epq (πq (x)) = πp (x) for p ≤ q, x ∈ A}. ←−

Clearly, lim A/Np (resp. lim Ap ) is a subalgebra of the product Πp∈sA ←−

←−

A/Np (resp. Πp∈sA Ap ) of normed (resp. Banach) algebras. lim A/Np ←−

(resp. lim Ap ) is called the projective limit of A/Np ’s (resp. Ap ’s). ←−

Theorem A.6.5. ([Mal86], p. 88; [Mi52], p. 13; [Zel73], p.33) Let (A, τ ) be a locally m-convex algebra whose topology τ is generated by a family sA = {p : p ∈ sA } of submultiplicative seminorms. Then e A ⊆ lim A/Np ⊆ lim Ap ⊆ A, ←−

←−

within topological algebraic isomorphism. If, in addition, A is complete, then A = lim A/Np = lim Ap , ←−

←−

within topological algebraic isomorphism. Proof. With the notations in the above definition, we can define ϕ : A → lim A/Np by ϕ(x) = (πp (x))p , i.e. the pth projection of ϕ(x) ←−

equals πp (x), x ∈ A. It is clear that ϕ(x) = 0 iff πp (x) = 0 for all p ∈ sA iff p(x) = 0 for all p ∈ sA iff x = 0, because A is Hausdorff. Hence ϕ is a one-one map. Further, since the topology τ ofQA is identical with the Cartesian product topology of its image ϕ(A) in A/Np , it follows that p∈sA Q ϕ is a homeomorphism of (A, τ ) onto the subalgebra ϕ(A) of A/Np p∈sA

and so A is isomorphic and homeomorphic to a subalgebra of a Cartesian

282


product of a family of normed algebras. Hence A can be identified with a subalgebra of Πp∈sA A/Np , the product of normed algebras. If A is metrizable, then the indexing set sA is countable and so each metrizable locally m-convex algebra is identifiable with a subalgebra of a countable product of normed algebras. Theorem A.6.6. ([Hus83], p. 11; [Zel73], p. 33) Let (A, τ ) be a locally m-convex F -algebra. Then A is isomorphic and homeomorphic to a closed subalgebra of Cartesian product of Banach algebras. Theorem A.6.7. ([Mal86], p. 85; [Hus83], p.10) Let (A, τ ) be a complete locally m-convex algebra whose topology τ is generated by a family sA of submultiplicative seminorms. Then: (a) σA (x) = ∪p∈sA σAp (xp ), x ∈ A. (b) rA (x) = supp∈sA rAp (xp ) = supp∈sA limn→∞ [p(xn )]1/n , x ∈ A. In the sequel, most of the results stated for locally m-convex algebras actually hold for the more general class of locally convex spaces having continuous inversion/ quasi-inversion (see [Mal86]). Theorem A.6.8. (1) ( [Mal86], p. 58; [Fra05], p. 46) If A is a locally m-convex algebra, then, for each x ∈ A, σA (x) is a non-empty (possibly non- compact) subset of C; hence the spectral radius rA (x) is finite if exists. (2) ( [Mal86], p. 60; [Fr05], p. 74) If A is a Q-algebra, then, for each x ∈ A, σA (x) is a (possibly empty) compact subset of C; hence the spectral radius rA (x) is finite if exists. (3) If A is a locally m-convex Q-algebra, then, for each x ∈ A, σA (x) is a non-empty compact subset of C ; hence rA (x) is finite and exists. Definition A.6.9. Let A be an algebra. A subalgebra I of A is called a left ideal (resp. right ideal ) if AI ⊆ I (resp. IA ⊆ I; it is called a two-sided ideal (or simply, an ideal ) if it is both a left and right ideal. A left (resp. right, two-sided) I in A is called a proper ideal if I 6= {0} and I 6= A. A proper left, right or two-sided ideal M in A is called a maximal left, right or two-sided ideal if, for any ideal (left, right or two-sided, respectively) L, M ⊆ L ⊆ A implies that L = M or L = A. Definition. ([Zel73], p. 27-28; [BD73], p.45) Let A be an algebra. A left (resp. right) ideal I in A is called a modular left (resp. modular


283

right) ideal if there exists an element u = urI (resp. u = ulI ) in A, called a modular right (resp. modular left) identity for A, such that xu − x ∈ I (resp. xu − x ∈ I) for all x ∈ A. A two-sided ideal I in A is called a modular ideal if there exists an element u = uI ∈ A, called a modular identity for A, such that ux − x ∈ I, xu − x ∈ I for all x ∈ A. Remarks. (1) ([Zel73], p. 27-28) A two-sided ideal I in A is modular iff A/I has an identity (the identity of A/I is the coset containing uI ). By Zorn’s lemma, each proper modular ideal is contained in a maximal modular ideal. Clearly, if an algebra A has already an identity e, then every two-sided ideal in A is a modular ideal with e a modular identity for A; hence in this case, every ideal in A is modular. (2) If A be a topological algebra, then the closure of every modular left (resp. right, two-sided) modular ideal in A is a modular left (resp. right, two-sided) ideal. Theorem A.6.10. ([Mal86], p. 67; [Fra05], p. 74-75) Every maximal modular left, right or two-sided ideal in a Q-algebra A is closed. In particular, every maximal ideal in a Q-algebra A with identity in A is closed. Proof. (see [Mal86], p. 67) We only prove for M a maximal modular two-sided ideal in A with modular identity u for M. By definition M is proper and u ∈ / M. Suppose M is not closed then M 6= M, and so, by maximality of M, M = A. Since A is a Q-algebra, Gq (A) is open and so u+ Gq (A) is an open neighborhood of u. Since u ∈ A = M , (u + Gq (A)) ∩ M 6= ∅. Let z ∈ (u + Gq (A)) ∩ M. Then there exist some w ∈ Gq (A), m ∈ M such that z = (u + w) = m, or u − m = −w ∈ Gq (A). Hence u − m has a q-inverse y (say) in A. Then (u −m) ◦ y = 0, or u −m+ y −(u −m)y = 0, or u −m+ y −uy + my = 0, or u = m + (uy − y) − my ∈ M + M + M = M. This is a contradiction since u ∈ / M. Hence M = M, and so M is closed. If A has an identity e, then each maximal ideal M in A is modular with modular identity e. Hence, by above argument, M is closed.

284


Note. ([Mal86], p. 67) In general, a maximal non-modular ideal in a locally m-convex algebra A may be dense in A and so need not be closed. Definition: (a) If A and B are two algebras, then a linear mapping ϕ : A → B is called an algebra homomorphism if ϕ(xy) = ϕ(x)ϕ(y) for all x, y ∈ A. In particular, a non-zero algebra homomorphism ϕ : A → K is called a multiplicative linear functional or a character on A. Note. (1) If ϕ is a multiplicative linear functional on an algebra A with identity e, then ϕ is non-zero iff ϕ(e) = 1. [Suppose ϕ is non-zero. Clearly, e2 = e implies [ϕ(e)]2 = ϕ(e) and so either ϕ(e) = 0 or ϕ(e) = 1. If ϕ(e) = 0, then, for any x ∈ A, ϕ(x) = ϕ(xe) = ϕ(x)ϕ(e) = 0; hence ϕ is identically zero, a contradiction.] (2) ([Mal86], p. 68) Any multiplicative linear functional ϕ on an algebra is onto. [Let λ ∈ C. Choose x ∈ A with ϕ(x) = α 6= 0. Then y = αλ x ∈ A with ϕ(y) = λ.] Recall that the co-dimension of a vector subspace M of a vector space E is defined as the dimension of its complementary subspace (or equivalently, the dimension of the quotient space E/M). Further, a vector subspace M of a vector space E is maximal iff M is of co-dimension one iff M = ϕ−1 (0) for some non-zero linear functional ϕ on E. Theorem A.6.11. Let A be an algebra. Then: (a) If ϕ is a multiplicative linear functional on A, then M = ϕ−1 (0) is a maximal modular ideal of co-dimension one in A. (b) If M ⊆ A is a maximal modular ideal of co-dimension one, then there exists a unique multiplicative linear functional ϕ on A such that M = ϕ−1 (0). (This need not hold if M is not of co-dimension one.) (c) There is a one-one correspondence between the set ∆(A) of multiplicative linear functional on A and the set M(A) of all maximal modular ideals of co-dimension one in A determined by parts (a) and (b). Proof. We give the proof only for the case of A an algebra with identity e. (a) If ϕ is a nonzero multiplicative linear functional, then it is easy to see that M = ϕ−1 (0) is a maximal ideal. To see that M is of co-dimension (i.e. A/ϕ−1 (0) is one-dimensional), let x ∈ A. Then ϕ(x −

ϕ(x) ϕ(x) e) = ϕ(x) − ϕ(x) = 0, or that x − e ∈ ϕ−1 (0) = M; ϕ(e) ϕ(e)


285

hence x∈M+

ϕ(x) e ⊆ M ⊕ Ce. ϕ(e)

Therefore, A = M ⊕ Ce (direct sum). Hence M = ϕ−1 (0) is a maximal ideal of codimension one. (b) Let M be a maximal ideal of A of codimension one. Then e ∈ /M because otherwise M = A, which contradicts the fact that M is a proper subset of A by definition. Hence A = M ⊕ Ce. Thus, for each x ∈ A, there is a unique λ ∈ C and m ∈ M with x = m + λe. Now define ϕ : A → C by ϕ(x) = λ for x = m + λe ∈ M ⊕ Ce = A. It is easy to verify that ϕ is a multiplicative linear functional with M = ϕ−1 (0). Further, ϕ(e) = 1, and so ϕ is nonzero. (c). This follows from (a) and (b). Note. (1) If M is a maximal ideal of an algebra A, then M need not be of co-dimension one; in fact, M may be of infinite co-dimension ([Zel73], p. 37, Exercise 8.4.a.). (2) However, if A is a complex Banach algebra, every maximal modular ideal M of A is automatically closed and of co-dimension one ([BD73], p. 79); also then the corresponding multiplicative linear functional ϕ is automatically continuous. Consequently, in this case, an analogue of the above theorem holds for commutative Banach algebras without mentioning the condition of co-dimension one. We now state two results on the automatic continuity of multiplicative linear functionals. Theorem A.6.12. ([BD73], p. 77; [Zel73], p. 37) Let ϕ be a multiplicative linear functional on a Banach algebra A. Then: (i) ϕ is continuous on A and ||ϕ|| ≤ 1. (ii) If A has an identity e with ||e|| = 1, then ||ϕ|| = 1. Theorem A.6.13. ([Mal86], p. 72-73) If A is a topological algebra with the property that every maximal modular two-sided ideal in A is closed, then every multiplicative linear functional ϕ on A is continuous. In particular, each multiplicative linear functional ϕ on a Q-algebra A is continuous. Remark.

286


([Zel73], p. 37) A multiplicative linear functional ϕ need not be continuous on a non-complete complex normed algebra A. Open Problem. As noted above, every multiplicative linear functional ϕ on a Banach algebra (or, more generally, on a Q-algebra) is continuous. In general, a multiplicative linear functional on a locally m-convex algebra A need not be continuous. It is still an open problem (known as the Michael-Mazur problem): whether every multiplicative linear functional on a commutative locally m-convex F -algebra A is continuous (see ([Mic52, Hus83, Dal00, Fra05]). Notation. The set of all multiplicative linear functionals on an algebra A is denoted by ∆(A). Clearly, ∆(A) ⊆ A′ , the algebraic dual of A, but ∆(A) need not be a vector subspace of A′ . If A is a topological algebra, let ∆c (A) denote the set of all (non-zero) continuous multiplicative linear functionals on A. In general, ∆(A) and ∆c (A) may be empty. For example, if A is the Arens algebra Lω [0, 1], then ∆(A) = ∅ (see [Zel73], p. 65). However, we have: Theorem A.6.14. ([Zel73], p. 38, 65; [Fra05], p. 55) If A is a commutative complete locally m-convex algebra (in particular, a commutative Banach algebra) with identity, then ∆c (A) 6= ∅. Theorem A.6.15. Let A be a commutative locally m-convex algebra. Then there is a one-one correspondence between the set ∆c (A) and the set Mc (A) of all closed maximal modular ideals of co-dimension one in A given by ϕ ↔ M, where M = ϕ−1 (0). Definition. Let A be an algebra. The intersection of all maximal modular left ideals of A is called the radical (or Jacobson radical) of A and is denoted by rad(A). Theorem A.6.16. ([Fra05], p. 60-61) Let A be an algebra. Then: (a) rad(A) is an ideal. (b) rad(A) ⊆ {x ∈ A : rA (x) = 0}. (c) If A is a commutative complete locally m-convex algebra, then rad(A) = {x ∈ A : rA (x) = 0}. Also, in this case, if the topology of A is generated by a family sA of continuous seminorms, rad(A) = {x ∈ A : lim [p(xn )]1/n = 0 for all p ∈ sA }. n→∞

(d) If A is a topological algebra with ∆c (A) 6= ∅, then rad(A) ⊆ ∩{ker(ϕ) : ϕ ∈ ∆c (A)} = {x ∈ A : ϕ(x) = 0 for all ϕ ∈ ∆c (A);


287

if, in addition, A is a commutative complete locally m-convex Q-algebra, rad(A) = ∩{ker(ϕ) : ϕ ∈ ∆c (A)}; hence, in this case, rad(A) is closed. Note. For a non-locally m-convex F -algebra A, rad(A) need not be closed. Definition. An algebra A is called semisimple if rad(A) = {0}; A is called simple if it has no two-sided ideal other than {0} and A. The quotient algebra A/rad(A) is semisimple. Every locally C ∗ -algebra and every topological algebra with an orthogonal basis is semisimple. (Recall that a topological basis {xn } for a topological algebra A is called orthogonal if xn xm = 0 if n 6= m and x2n = xn [Hus83, p. 62]. If A is a commutative complete locally m-convex Q-algebra with ∆c (A) 6= ∅, then, by above theorem, A is semisimple iff {x ∈ A : ϕ(x) = 0 for all ϕ ∈ ∆c (A)} = {0}. Definition: Let A be a topological algebra with ∆c (A) 6= ∅, and let w(A∗ , A) be the weak∗ -topology of A∗ . Then the relative weak∗ topologies s = w(A∗ , A)|∆c (A) of A∗ on ∆c (A) is called the Gelfand topology. ∆c (A) with this topology is called the maximal ideal spaces of A (or the topological spectrum of A) and is denoted by (∆c (A), s). In general, (∆c (A), s) need not be compact or even locally compact. Recall that, if A is a TVS, the every equicontinuous subset of A∗ is relatively w(A∗ , A)-compact (Alaoglu-Bourbaki theorem; see [Hor66], p. 201). Theorem A.6.17. Let A be a topological algebra. Then: (a) ( [Mal86], p. 140) (∆c (A), s) is a completely regular Hausdorff space. (b) ([Mal86], p. 75) If A is a Q-algebra, then (∆c (A), s) is equicontinuous and hence locally compact. (c) ([Mal86], p. 143) If A is an F -algebra, then (∆c (A), s) is compact iff A is a Q-algebra. (d) ([Hus83], p. 12) If A is a complete locally m-convex algebra, then ∆c (A) ∪ {0} is a w(A∗, A)-closed subset of A∗ . (e) ([Hus83], p. 12) If A is a complete locally m-convex algebra with identity, then 0 is an isolated point of ∆c (A) ∪ {0} and ∆c (A) is a w(A∗, A)-closed subset of A∗ . Definition A.6.18. ([Mal86], p. 142)

288


Let A be a topological algebra. Then ∆c (A) is called locally equicontinuous, as a subset of A∗ if, for each f ∈ ∆c (A), there exists a neighborhood U of f in ∆c (A) such that U is an equicontinuous subset of (∆c (A), s) ⊆ (A∗ , w(A∗ , A)). Definition A.6.19. ([Mal86], p.73) Let A be a topological algebra. For each x ∈ A, define xˆ : ∆c (A) → C by xˆ(ϕ) = ϕ(x), ϕ ∈ ∆c (A). Then xˆ is continuous on ∆c (A); hence xˆ ∈ C(∆c (A)) = C(∆c (A), s)). The function xˆ is called the Gelfand transform of x on A. Setting Aˆ = {ˆ x : x ∈ A}, we have Aˆ ⊆ C(∆c (A)). The map G : x → xˆ is called the Gelfand representation of A into C(∆c (A)). Theorem A.6.20. ([Mal86], p. 143) Let A be a topological algebra. Then: (a) (∆c (A), s) is locally equicontinuous iff ∆c (A) is locally compact b ⊆ C(∆c (A)) is continuous. and G : A → A (b) If A is a locally m-convex Q-algebra and (∆c (A), s) is locally compact, then ∆c (A) is locally equicontinuous. Theorem A.6.21. ([Mal86], p. 140, 266) Let A be a topological algebra. Then: (a) The map G : x → xˆ is a continuous algebraic homomorphism of A onto the subalgebra Aˆ of C(∆c (A)). (b) G is one-one iff A is semisimple. The commutative version of the Gelfand-Naimark theorem for topological algebras is as follows. Theorem A.6.22. (Gelfand-Naimark) ([Mic52], p. 34-35; [Mal86], p. 488) (a) If A is a commutative complete semisimple locally m-convex algebra, then A is both algebraically and topologically isomorphic to a separating subalgebra Aˆ of (C(△c (A), k), k being the compact-open topology; if A has an identity e, then Aˆ contains the identity function. (b) If A is a commutative locally C ∗ -algebra and that the Gelfand map G : x → xˆ is continuous, then A is algebraically ∗-isomorphic to + (Co (△+ c (A), k), where △c (A) = △c (A) ∪ {0}. Definition. Let A be a topological algebra.


289

(1) A net {eλ : λ ∈ I} in A is called a left ( resp. right) approximate identity of A if lim eλ x = x (resp. lim xeλ = x) for all x ∈ A. λ

λ

(2) {eλ : λ ∈ I} is called a two-sided approximate identity (or an approximate identity) of A if it is both a left and right approximate identity. Every locally C ∗ -algebra A has a uniformly bounded approximate identity {eλ : λ ∈ I}. Definition. An algebra A is called (a) left faithful if, for any a ∈ A, aA = {0} ⇒ a = 0. (b) right faithful if, for any a ∈ A, Aa = {0} ⇒ a = 0. (c) faithful if it is both left and right faithful. Other equivalent terms for “faithful ” are “proper ”, “without order ”, “without annihilators”. Theorem A.6.23. An algebra A is faithful in each of the following cases: (a) A has an identity e. (b) A is a topological algebra having an approximate identity {eλ : λ ∈ I} (e.g. A is a C ∗ -algebra or locally C ∗ -algebra). (c) A is a topological algebra with an orthogonal basis. (d) A is a commutative semisimple Banach algebra. Proof. (a) This is clear. (b) Suppose A is a topological algebra with an approximate identity {eλ : λ ∈ I}. Let a ∈ A with ax = 0 for all x ∈ A. Then aeλ = 0 for all λ ∈ I; hence a = lim aeλ = 0. Similarly, Aa = {0} ⇒ a = 0. λ

(c) Suppose A is a topological algebra with an orthogonal basis {xn }. P Let a ∈ A with ax = 0 for all x ∈ A. We can write a = n≥1 tn xn . Then, since each xm ∈ A, X axm = 0, or tn (xn xm ) = 0, or tm x2m = 0, m≥1

P and so tm = 0 for all m ≥ 1. Hence a = m≥1 tm em = 0. Similarly, Aa = {0} ⇒ a = 0. (d) Suppose A is a semisimple. Let a ∈ A with ax = 0 for all x ∈ A. Then, a.a = 0 and so,by induction, an = 0 for all n ≥ 1. Hence

290

A. TOPOLOGY AND FUNCTIONAL ANALYSIS 1

lim kan k n = 0,showing that

n→∞

1

a ∈ {x ∈ A : lim kxn k n = 0} = rad(A). n→∞

Since A is semisimple, rad(A) = {0} and so a = 0.

Example. Let A be any vector space (resp. TVS) over C with A 6= {0}. Then A becomes a commutative algebra (resp. topological algebra) without an identity under the trivial multiplication: x.y = 0 for all x, y ∈ A. Then, for any non-zero x ∈ A, xA = {0}. However x 6= 0, and so A is not faithful. Recall that if E is a TVS, then CL(E) = CL(E, E) is an algebra under composition and has identity I : E → E given by I(x) = x (x ∈ E). Theorem A.6.24. Let E be a TVS (in particular, a topological algebra). Then (CL(E), tu ) and (CL(E), tp ) are topological algebras. Proof. We need to show that the composition map (S, T ) → ST is separately continuous. Fix So ∈ CL(E), and let {Tα : α ∈ I} ⊆ CL(E) tu tu with Tα −→ T ∈ CL(E). To show that So Tα −→ So T , let U(D, W ) be any tu -neighborhood of 0 in CL(E). By continuity of So : E → E, there tu exists V ∈ WE such that So (V ) ⊆ W. Since Tα −→ T , there exists αo ∈ I such that (Tα − T )(D) ⊆ V for all α ≥ αo . Then, for any α ≥ αo , So (Tα − T )(D) ⊆ So (V ) ⊆ W. t

u Hence So Tα −→ So T. tu Next, fix To ∈ CL(E), and let {Sα : α ∈ I} ⊆ CL(E) with Sα −→ tu S ∈ CL(E). To show Sα To −→ STo , let U(D, W ) be any uneighborhood of 0 in CL(E). Since To : E → E is continuous, To is bounded and so To (D) is a bounded set in E. Then U(To (D), W ) is a tu -neighborhood of tu 0 in CL(E). Since Sα −→ S, there exists αo ∈ I such that

Sα − S ∈ U(To (D), W ) or (Sα − S)To (D) ⊆ W for all α ≥ αo . t

u Hence Sα To −→ STo . Thus (CL(E), tu ) is a topological algebra. By the same argument, (CL(E), tp ) is also a topological algebra. Recall that a subcollection B of the collection b(E) of all bounded subsets of a TVS E is called fundamental system of bounded sets if, for each bounded set D ⊆ E, there exists a Di ∈ B such that D ⊆ D1 ([Hus65], p. 20).


291

Theorem A.6.25. Let E be an F-space (in particular, F-algebra). If E has a countable fundamental system of bounded sets, then (CL(E), tu ) is a complete metrizable topological algebra with jointly continuous multiplication. Proof. Let D = {D1 , D2 , ....} be a countable fundamental system of bounded sets in E. Since E is metrizable, it has a countable base WE = {W1 , W2 , ...} (say) of neighborhoods of 0. Then clearly the collection {U(Di , Wj ) : i, j = 1, 2, 3, ....} is a countable base of neighborhoods of 0 in (CL(E), tu ). Also (CL(E), tu ) is Hausdorff. Hence, by the metrization theorem of TVSs, (CL(E), tu ) is metrizable. Therefore (CL(E), tu ) is a complete metrizable topological algebra. Thus, by the Aren’s theorem, (CL(E), tu ) is a topological algebra with jointly continuous multiplication. Definition: Let (A, q) be an F -normed algebra. For any T ∈ CL(A), let kT kq =

sup

q(T (x)).

(∗)

x∈A,q(x)≤1

In general, kT kq need not exist since the set {x ∈ A : q(x) ≤ 1} may not be bounded in the F -algebra (A, q). However, for q a k-norm (0 < k ≤ 1), the existence and other useful properties of ||.||q are summarized in the following theorem (see [Rol85], p. 101-102; [Bay03], p. 3-5; [ARK11]). Theorem A.6.26. Let (A, q) be a k-normed algebra, where 0 < k ≤ 1. Then: (a) A linear mapping T : A → A is continuous ⇔ ||T ||q < ∞. (b) k.kq is a k-norm on CL(A). (c) For any T ∈ CL(A), kT kq = sup

x∈A,x6=0

q(T x)) . q(x)

(d) For any T ∈ CL(A), q(T x) ≤ kT kq .q(x) for all x ∈ A. (e) For any S, T ∈ CL(A), kST kq ≤ kSkq kT kq ; hence (CL(A), k.kq ) is an k-normed algebra. (f) If A is complete, then (CL(A), k.kq ) is also complete. (g) If A has a minimal approximate identity {eλ : λ ∈ I}, then, for any a ∈ A, kLa kq = kRa kq = q(a), where La , Ra : A → A are the maps given by La (x) = ax and Ra (x) = xa, x ∈ A.

292


Proof. (a) This is clear. (b) Clearly kT kq ≥ 0 and kT kq = 0 iff T = 0. For any T ∈ CL(A) and λ ∈ K, q(λT (x)) = |λ|k sup q(T (x)) = |λ|k kT kq . kλT kq = sup q(λT (x)) q(x) q(x)≤1 q(x)≤1 It is easy to verify that, for any S, T ∈ CL(A), kS + T kq ≤ kSkq + kT kq . (c) For any x (6= 0) ∈ A, if y = x 1 , then [q(x)] k

q(y) = q[

x q(x)

1 k

1

]=

1

[q(x) k ]k

q(x) = 1;

hence sup x∈A,x6=0

q(T x)) = q(x)

q[T ( sup

1

x 1 q(x) k

.q(x) k )]

q(x)

x∈A,x6=0 1 k

[q(x) ]k .q[T ( =

sup sup q[T ( x∈A,x6=0

1

)]

q(x)

x∈A,x6=0

=

x q(x) k

x 1

q(x) k

)] =

sup y∈A,q(y)=1

q[T (y)] = kT kq .

(d) Let T ∈ CL(A). Using (c), clearly q(T x) ≤ kT kq .q(x) for each x 6= 0 in A Clearly, this also holds for x = 0. (e) Let S, T ∈ CL(A). Using (d), kST kq = ≤

sup q[S(T (x))] ≤ sup kSkq q(T (x))

q(x)≤1

q(x)≤1

sup kSkq kT kq · q(x) = kSkq . kT kq .

q(x)≤1

(f) By Theorem A.6.25, (CL(A), k.kq ) is complete and so it is an F -algebra. See also ([Bay03], p. 5; [Rol85], p. 41). (g) Let a ∈ A. Then kLa kq = sup q(La (b)) = sup q(ab)) ≤ sup q(a).q(b) = q(a). q(b)≤1

q(b)≤1

q(b)≤1

On the other hand, since q(eλ ) ≤ 1 for all λ ∈ I, kLa kq = sup q(ab)) ≥ q(aeλ ) for all λ ∈ I; q(b)≤1

so kLa kq ≥ lim q(aeλ ) = q(lim aeλ ) = q(a). λ

λ

HencekLa kq = q(a). Similarly, kRa kq = q(a).


293

Remark. In the definition of kT kq given by (∗), we cannot make the blank assumption that, for q an F -norm on E, kT kq always exists for each T ∈ CL(E). This assumption cannot be justified in view of the following counter-examples. (1) First, ||T ||q need not be finite for a general F-norm. For example, let E = R2 ; q(x1 , x2 ) = |x1 | + |x2 |1/2 ; T (x1 , x2 ) = (x2 , x1 ). Then q is an F -norm on E, but ||T ||q = ∞: for any n ∈ N, ||T ||q ≥

q(n2 , n) n2 + n1/2 q[T (n, n2 )] = = → ∞. q(n, n2 ) q(n, n2 ) 2n

(2) Even when considering the subspace of those T for which ||T ||q < ∞, then ||.||q need not always be an F-norm, since (F5 ) need not hold. For example, for a fixed sequence (pn ) with 0 < pn ≤ 1, pn → 0, consider the F-algebra E of sequences (xn ) ⊆ R with |xn |pn → 0 and q((xn )) = supn≥1|xn |pn . Then ||T ||q < ∞ for all multipliers of E, but ||.||q is not an F -norm, i.e. it makes the space CL(E) into an additive topological group but not into a topological vector space (as it would lack the continuity of scalar multiplication in the absence of (F5 ) (cf. [Rol85], Example 1.2.3, p. 8)). In view of the above remark, we shall need to assume that (E, q) is a k-normed space (or a k-normed algebra) whenever ||T ||q is considered for T ∈ CL(E). Theorem A.6.27. Let (A, q) be an k-normed algebra with q submultiplicative and A having an identity e. Then A is isometrically isomorphic to a closed subalgebra of (CL(A), tu ). Proof. Define q / : A −→ R by q(xz) , x∈A z∈A,z6=0 q(z)

q / (x) = sup Then, for any x ∈ A, q / (x) = sup z∈A z6=0

q(xz) q(x)q(z) ≤ sup = q(x), q(z) q(z) z∈A,z6=0

and also q(x) = q(xe) = q(e). Hence, q / is equivalent to q.

q(xe) q(xz) ≤ q(e) sup = q(e)q / (x). q(e) q(z) z6=0

294


Now for each x ∈ A, consider Lx : A −→ A given by Lx (y) = xy, y ∈ A. Then clearly {Lx : x ∈ A} is a subalgebra of CL(A). Further, q(xz) q(Lx (z)) = sup = q / (x). q(z) q(z) z∈A,z6=0 z∈A,z6=0

kLx kq = sup

Hence (A, q / ) is isometrically isomorphic to the subalgebra {Lx : x ∈ A } of CL(A) under the mapping x → Lx . Next {Lx : x ∈ A} is closed in (CL(E), tu ), as follows. Let T ∈ {Lx : x ∈ A}

k.kq

. Then there exists a sequence {Lxn } ⊆ {Lx : x ∈ A}

k. kq

such that Lxn −→ T . We show that T = Lxo for some xo ∈ A. Now, for any y ∈ A, T (y) = =

lim Lxn (y) = lim xn y = lim (xn e)y

n→∞

n−→∞

n→∞

lim Lxn (e)y = T (e)y = LT (e) (y);

n→∞

i.e. T = Lxo , where xo = T (e) ∈ A. Hence T ∈ {Lx : x ∈ A}, and so {Lx : x ∈ A} is closed in (CL(E), tu ). Definition. Let Y be a TVS and A a topological algebra, both over the same field K (= R or C). Then Y is called a topological left A-module if it is a left A-module and the module multiplication (a, x) → a.x from A × Y into Y is separately continuous. Definition: Let Y be a topological left A-module, and let b(A) (resp. b(Y )) denote the collection of all bounded sets in A (resp. in Y ). Then: (a) The module multiplication (a, x) → ax is called b(A)- (resp. b(Y )) hypocontinuous if, given any neighborhood G of 0 in Y and any D ∈ b(A) (resp. B ∈ b(Y )), there exists a neighborhood H of 0 in Y (resp. V of 0 in A) such that DH ⊆ G (resp. V B ⊆ G). (b) The module multiplication (a, x) → ax is called jointly continuous if, given any neighborhood G of 0 in Y , there exist neighborhoods U of 0 in A and H of 0 in Y such that UH ⊆ G. Clearly, joint continuity ⇒ hypocontinuity ⇒ separate continuity. The following theorem shows that, under some additional conditions, separate continuity ⇒ hypocontinuity. Theorem A.6.28. Let (Y, τ ) be a topological left A-module. If Y is ultrabarrelled, then the module multiplication is b(A)-hypocontinuous. In particular, every ultrabarrelled topological algebra has a hypocontinuous multiplication.


295

Proof. Let G be a neighborhood of 0 in Y and D ∈ b(A). For any a ∈ A, define La : Y → Y by La (x) = a.x, x ∈ Y . Clearly, each La is linear and also continuous (by separate continuity of the module multiplication). Further, {La : a ∈ D} is pointwise bounded in CL(Y, Y ). [Let x ∈ Y and G1 a neighborhood of 0 in Y . Since D in bounded in A, by separate continuity of the module multiplication, D.x is bounded in Y and so there exists r > 0 such that D.x ⊆ rG1 . So {La (x) : a ∈ D} = {a.x : a ∈ D} = D.x ⊆ rG1 , showing that {La : a ∈ D} is pointwise bounded in CL(Y, Y ).] Since Y is ultrabarralled, by the principle of uniform boundedness, {La : a ∈ D} is equicontinuous. Hence, given any G ∈ WY , there exists a neighborhood H of 0 in Y such that La (H) ⊆ G for all a ∈ D; i.e. D.H ⊆ G.

296


7. Measure Theory In this section, we present some measure theoretic terminology (see [Hal50, Var65, Whe83, Bog07]). Definition. Let X be a given set. A collection A(X) of subsets of X is called an algebra if it satisfies the following conditions: (a) X ∈ A(X); (b) The union of finite number of sets in A(X) is also in A(X); (c) For any A, B ∈ A(X), the difference set A\B is also in A(X). Clearly ∅ ∈ A(X). Further, A(X) is closed under finite intersections. An algebra A(X) is called a σ-algebra if it closed under countable unions (hence also under countable intersections). Definition. If A(X) is an algebra of subsets of X, then a function µ : A(X) → R+ is called (i) finitely additive if, for any A1 , ..., An ∈ A(X) with Ai ∩ Aj = ∅ for i 6= j, n X µ(∪ni=1 Ai ) = µ(Ai ); i=1

(ii) countably additive if, for any sequence of sets A1 , A2 , ..... ∈ A(X) S∞ with i=1 Ai ∈ A(X) and Ai ∩ Aj = ∅ for i 6= j, µ(∪∞ i=1 Ai )

=

∞ X

µ(Ai ).

i=1

Now, let X be a completely regular Hausdorff space, and let c(X) (resp. k(X), z(X), cz(X)) the collection of all closed (resp. compact, zero, cozero) subsets of X (see Section A.1). We refer to [GJ60] for general facts about z(X) and cz(X). The family of zero sets is preserved by finite unions and countable intersections. A compact Gδ -set is a zero set. Any zero set is closed and a Gδ -set; the converse is true in a normal space. The family of cozero sets is preserved by finite intersections and countable unions, and forms a base for the topology of X. Disjoint zero sets are contained in disjoint cozero sets. Definition. Let Ba(X) denote the smallest σ-algebra of subsets of X containing z(X), and let Bo(X) denote the smallest σ-algebra of subsets of X containing c(X). Clearly, the σ-algebras Ba(X) and Bo(X) are closed under countable unions and intersections The members of Ba(X) (resp. Bo(X)) are called Baire (resp. Borel ) sets. Since every zero set

7. MEASURE THEORY

297

is closed, Ba(X) ⊆ Bo(X), but, in general, Ba(X) 6= Bo(X) even if X is compact. For instance, in the compact Hausdorff space [0, 1]ω1 , singleton sets are closed sets but are not zero sets. It is true that every compact set in the Baire σ-algebra Ba(X) is a zero set, but not every compact K ⊆ X is in Ba(X). See ([Whe83], p. 109). However, if X is a metric space or, more generally, a perfectly normal space, then Ba(X) = Bo(X) ([Whe83], p. 109; [Bog07], p. 13). (A topological space X is called perfectly normal if every closed set F ⊆ X has the form Z = f −1 (0) for some f ∈ C(X), or equivalently, if it is normal and every closed subset of X is a Gδ -set.) Definition. If Σ is a subfamily of Ba(X), then we say that a finite, real-valued, finitely additive set function µ on Ba(X) is Σ-regular if, for each A ∈ Ba(X) and ε > 0, there exists F ∈ Σ, F ⊆ A, such that |µ(H)| < ε for all H ∈ Σ with H ⊆ A\F. A positive Baire measure µ on X is a finite, real-valued, non-negative, finitely additive set function on the algebra Ba(X) which is z(X)-regular (i.e., equivalently, for any B ∈ Ba(X), µ(B) = sup{µ(Z) : Z ⊆ B, Z ∈ z(X)}. A (signed) Baire measure is the difference of two positive Baire measures. The spaces of all Baire measures and positive Baire measures are denoted by M(Ba(X)) and M + (Ba(X)), respectively. If µ ∈ M(Ba(X)), then the set functions µ+ and µ− defined on Ba(X) defined by µ+ (A) = sup{µ(B) : B ∈ Ba∗ (X), B ⊆ A}, µ− (A) = − inf{µ(B) : B ∈ Ba∗ (X), B ⊆ A}, are members of M + (Ba(X)) and µ = µ+ − µ− . The set function |µ| = µ+ + µ− is again in M + (Ba(X)) and is called the total variation of µ. If A ∈ Ba(X), |µ|(A) = sup{

n X i=1

∗

|µ(Bi )| : Bi ∈ Ba (X), A =

n [

Bi }.

i=1

The inner and outer inner measures determined by µ ∈ M + (Ba(X)) are µ∗ (A) = sup{µ(Z) : Z ∈ z(X), Z ⊆ A}, µ∗ (A) = inf{µ(U) : U ∈ cz(X), A ⊆ U}, for each subset A of X.

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M(Ba(X)) is a Banach space with the norm ||µ|| = |µ|(X) and M (Ba(X)) is a closed subset of M(Ba(X)). +

Definition. If µ ∈ M + (Ba(X)) and f ∈ Cb (X). If f ≥ 0, we define the integral Z Z f dµ = sup{ hdµ : 0 ≤ ϕ ≤ f, ϕ a simple function w.r.t. Bo(X). X

X

For arbitrary f ∈ Cb (X), we can write f = f + − f − , where f + (x) = max{f (x), 0} and f − (x) = max{−f (x), 0}. Then we define Z Z Z + f dµ = f dµ − f − dµ. X

X +

X

Now if µ ∈ M(Ba(X)), µ = µ − µ with µ+ , µ− ∈ M + (Ba(X)), and f ∈ Cb (X), we define Z Z Z + f dµ = f dµ − f dµ− . X

X

−

X

For a full discussion of integration with respect to a finitely additive measures, see [Hal50, Roy88, DS58]. The following version of the classical Riesz representation theorem is well-known (see, [Var65, Whe83]). Theorem A.7.1 (Riesz-Alexandroff) Let X be a completely regular Hausdorff space. Then (Cb (X), u)∗ = M(Ba(X)) via the linear isomorphism L ←→ µ, where Z L(f ) = f dµ for all g ∈ Cb (X) and ||L|| = |µ|(X). x

Remark. ([Whe83], p. 117) The Borel measures µ on Bo(X) arise in the usual Riesz Representation Theorem for compact spaces. For a short and elegant proof of this result, see Garling [Gar73]. In the completely regular space, several difficulties arise if we attempt to represent ∗ members of (Cb (X), u) by Borel measures. Each L ∈ (Cb (X), u)∗ can be represented by a finitely additive c(X)-regular measure ν on Bo(X), but ν need not be unique; unfortunately, there may be uncountably many ν which work. Uniqueness of the representing Borel measure is restored if X is normal. ∗ In view of the Riesz-Aleksandrov theorem, any L ∈ (Cb (X), u) R can be identified with a unique µ ∈ M(Ba(X)) and may write µ(f ) for x f dµ for brevity. In his fundamental paper, Varadarajan [Var65] analyzed three subspaces of M(Ba(X)), as follows:

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Definition. (1) A measure µ ∈ M(Ba(X)) is called σ-additive if, for any sequence {Zn } of zero sets of X with Zn ↓ ∅, |µ(Zn )| → 0. It is equivalent to that µ be countably additive on Ba(X), so that µ can be extended uniquely to a countably additive set function on Ba(X). We shall always assume that this extension has been made. Any countably additive measure on Ba(X) must be z(X)-regular (using the fact that every cozero set is a countable union of zero sets), hence is σ-additive. (2) µ is called τ -additive if it is c(X)-regular and, for any net {Zα } of zero sets of X with Zα ↓ ∅, |µ(Zα)| → 0. This behavior is not preserved by decreasing net of cozero sets. If X is locally compact, every τ -additive measure is compact zero set regular. Any τ -additive measure µ on Ba(X) can be extended to a unique c(X)-regular Borel measure ν such that Fα ↓ F , ν(Fα ) → ν(F ). If every σ-additive measure in M(Ba(X)) is τ -additive, then X is called measure compact. (3) µ is called tight if, given any ε > 0, there exits a compact set K ⊆ X such that |µ|(X) < |µ|∗ (K) + ε. Any tight Baire measure can be extended to a k(X)-regular Borel measure ν. We shall always assume that this extension has been made. Let Mσ (Ba(X)) (resp. Mτ (Bo(X)), Mt (Bo(X))) denote the space of all bounded real-valued σ-additive (resp. τ -additive, tight) measures on Ba(X) (resp. Bo(X), Bo(X)). Restricting the above measures to Ba(X), we clearly have Mt (Ba(X)) ⊆ Mτ (Ba(X)) ⊆ Mσ (Ba(X)) ⊆ M(Ba(X)). Note that Mt (Bo(X)) ⊆ Mσ (Ba(X)) need not hold in general as an µ ∈ Mτ (Bo(X)) is not necessarily z(X)-regular. Every tight µ ∈ M(Ba(X)) is τ -additive and hence it has a unique extension µ′ ∈ Mτ (Bo(X)).

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8. Uniform Spaces and Topological Groups Definition. [Will70, p. 238] Let Y be a non-empty set. (1) The subset ∆ = ∆(Y ) = {(x, x) : x ∈ Y } of Y × Y is called the diagonal on Y . (2) For any A ⊆ Y × Y, we define we define A−1 by A−1 = {(y, x) : (x, y) ∈ A}. If A = A−1 , then A is called symmetric. (3) For any A, B ⊆ Y × Y, we define A ◦ B by A ◦ B = {(x, y) ∈ Y × Y : ∃ z ∈ Y such that (x, z) ∈ A and (z, y) ∈ B}. If A = B, we shall write A ◦ A = A2 . Note that, (x, z) ∈ A, (z, y) ∈ B ⇒ (x, y) ∈ A ◦ B. Definition. [Will70, p. 238] A subset H of subsets of Y × Y is called a uniformity (or a uniform structure) on Y if (u1 ) △(Y ) = {(x, x) : x ∈ Y } ⊆ A for all A ∈ H. (u2 ) A ∈ H =⇒ A−1 ∈ H, i.e. each A ∈ H is symmetric. (u3 ) If A ∈ H, ∃ some B ∈ H such that B 2 ⊆ A. (u4 ) If A, B ∈ H, ∃ some C ∈ H such that. C ⊆ A ∩ B. (u5 ) If A ∈ H and A ⊆ B, then B ∈ H. In this case, the pair (Y, H) is called a uniform space and we shall often denote it by H(Y ). The members of H are called vicinities of H. Definition. [Will70, p. 238-239] (1) Let H1 and H2 be two uniformities on a set Y . If H1 ⊆ H2 , then H1 is said to be weaker than H2 and H2 is said to be stronger than H1 . (2) The weakest uniformity on a set Y is I = {Y × Y }, called the indiscrete uniformity. (3) The strongest uniformity on a set Y is D = {A ⊆ Y ×Y : ∆ ⊆ A}, called the discrete uniformity. Definition: Let H be a uniformity on a set Y . Then a subcollection B ⊆ H is called a base ( or basis) for H if, for any A ∈ H, there exists B ∈ B such that B ⊆ A. Theorem A.8.1. Let B be a base for a uniformity H on Y . Then it has the following properties: (b1 ) △ ⊆ B for all B ∈ B : (b2 ) If B ∈ B, ∃ some C ∈ B such that C −1 ⊆ B. (b3 ) If B ∈ B, ∃ some C ∈ B such that C 2 ⊆ B.

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301

(b4 ) If A, B ∈ B, ∃ some C ∈ B such that C ⊆ A ∩ B. Conversely, if a collection B of subsets of Y × Y satisfies the above conditions (b 1 )-(b 4 ), then the collection H = {A ⊆ Y × Y : A contains some B ∈ B} is a uniformity on Y whose base is B. We next define a topology on a set Y induced by a uniformity H on Y. Definition: [Edw65, p. 26] Let H be a uniformity on a set Y . For any x ∈ Y and A ∈ H, let NA (x) = {y ∈ Y : (x, y) ∈ A}. Then, for each x ∈ Y , the collection {NA (x) : A ∈ H} satisfies all the conditions of a neighborhood system at x. Consequently, there exists a unique topology τH on Y such that, for each x ∈ Y , the collection {NA (x) : A ∈ H} forms a local base at x. In this case, the topology τH on Y is said to be the topology induced by (or derived from) the uniformity H. Notation. For any K ⊆ Y and A ∈ H, let NA (K) = ∪{NA (x) : x ∈ K} = {y ∈ Y : (x, y) ∈ A for some x ∈ K}. As we shall see below, every semi-metric (in particular, metric) space is a uniform space and every uniform space is a topological space. Theorem A.8.2. [Edw65, p. 30] Let (Y, d) be a semi-metric space (in particular, a metric space). For each r > 0, let Dr = {(x, y) ∈ Y × Y : d(x, y) < r}. Then the collection B = {Dr : r > 0} is a base for a uniformity on Y . More generally, we have: Theorem A.8.3. [Edw65, p. 27] Let D = {dα : α ∈ I} be a family of semi-metrics on a set Y , and let D + = {max{dα1 , ..., dαn } for all finite {α1 , ...αn } ⊆ I}. For each r > 0, let Dd,r = {(x, y) ∈ Y × Y : d(x, y) < r}, d ∈ D + . Then the collection B = {Dd,r : d ∈ D + , r > 0} is a base for a uniformity on Y . (Note that D ⊆ D + .) Definition: [Edw65, p. 28]

302


A topological space (Y ,τ ) is said to uniformizable if its topology can be derived from a uniformity, i.e., there exists a uniformity H on Y such that τ = τH . Theorem A.8.4. (a) [Will70, p. 256] A topological space (Y, τ ) is uniformizable ⇔ it is completely regular. (b) [Will70, p. 257] A uniformity H on Y is pseudometrizable ⇔ it has a countable base. (b) [Edw65, p. 27-28] A topological space (Y, τ ) is uniformizable with uniformity H ⇔ there exists a family {dα : α ∈ I} of semi-metrics on Y , which generates the uniformity H. Note. In particular, every metric space and every completely regular space is a uniform space. Definition: [Edw65, p. 28] Let (Y, H) be uniform space. (1) A net {xα : α ∈ I} in Y is said to be convergent to x ∈ Y if, given any A ∈ H, there exists an α0 = α0 (A) ∈ I such that (xα , x) ∈ A for all α ≥ α0 ; in this case, we write xα → x. (2) A net {xα : α ∈ I} in Y is said to be a Cauchy net if, given any A ∈ H, there exists an α0 = α0 (A) ∈ I such that (xα , xβ ) ∈ A for all α, β ≥ α0 . Definition: A uniform space (Y, H) is said to be complete if every Cauchy net in Y is convergent to a point in Y. Definition: [Edw65, p. 30] Let (Y, H) be a uniform space. The set S ⊆ Y is said to be totally bounded if, given any A ∈ H, there exists a finite set {x1 , x2 , ...., xn } ⊆ Y such that S ⊆ ∪ni=1 NA (xi ) = ∪ni=1 {y ∈ Y : (y, xi ) ∈ A}. Equivalently, [Will70, p. 262], a set S ⊆ Y is totally bounded if, given any A ∈ H, there exists a finite cover {U1 , U2 , ...., Un } of S such that {∪ni=1 Ui × Ui ⊆ A}.


303

Theorem A.8.5 [Will70, p. 262] A subset S of a uniform space (Y, H) is compact ⇔ it is complete and totally bounded. Definition. [Will70, p. 243] (a) Let (Y, HY ) and (Z, HZ ) be uniform spaces and xo ⊆ Y. A function f : Y → Z is said to be continuous at xo if, for any B ∈ HZ , there exists an A ∈ HY such that (f (x), f (xo )) ∈ B whenever x ∈ Y with (x, xo ) ∈ A. (b) Let (Y, HY ) and (Z, HZ ) be uniform spaces and S ⊆ Y. A function f : Y → Z is said to be uniformly continuous on S if, for any B ∈ HZ , there exists an A ∈ HY such that (f (x′ ), f (x′′ )) ∈ B whenever x′ , x′′ ∈ Y with (x′ , x′′ ) ∈ A. Note. If f is uniformly continuous on S ⊆ Y, then clearly f is continuous on S; the converse holds if S is compact. We next recall the notion of a ”topological group” and show these spaces are uniform spaces. Notations: Let G = (G, ·) be a group. For any V, W ⊆ G, let V · W = {a · b : a ∈ V, b ∈ W }, V −1 = {a−1 : a ∈ V }. Definition: Let G = (G, ·) be a group, written multiplicatively, and τ a topology on G. Then the G is called a topological group if (TG1 ) The mapping P : (x, y) → x · y of G × G → G is jointly continuous (i.e. given any x, y ∈ G and any neighborhood U of xy, there exist neighborhoods V and W of x and y respectively such that P (V, W ) ⊆ U or V · W ⊆ U). (TG2 ) The inverse mapping I : x → x−1 of G → G is continuous (i.e. given any x ∈ G and any neighborhood U of x−1 , there exists a neighborhood V of x such that I(V ) ⊆ U or V −1 ⊆ U). In this case, we denote G = (G, ·, τ ). Note. The two conditions (TG1 & TG2) are equivalent to the single condition: (TG) The map (x, y) → xy −1 of G × G → G is jointly continuous. Examples: (1) (R, +) with the usual topology is a TG. (2) (R\{0}, ·) with the usual topology is a TG.

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(3) (Q, +), (Q\{0}, ·) with the induced topologies are topological groups. (4) (C, +), (C\{0}, ·) are TG. (5) Every TVS (X, τ ) is a TG under the operation of +; in fact, by (TVS1 ), (x, y) → x + y is continuous and, by (TVS2 ), the inversion x → x−1 = −x is continuous. In particular, every normed vector space (X, k·k) is a TG under the operation of +. (6) T = {λ ∈ C: |λ| = 1} is a TG under multiplication; here λ−1 = λ. It is called a torus group [Pa78, p. 88]. Remark. By definition, any TVS (E, τ ) is topological group under +. Lemma A.8.6. Any group G = (G, ·) with the discrete topology τI is a TG. Proof. (TG1 ) We verify that the map (x, y) → x · y is continuous. Let x, y ∈ G. Let U be any neighborhood of xy in (G, τI ). Since (G, τI ) is a discrete space V = {x} and W = {y} are τI -open neighborhoods of x and y, respectively. Clearly V · W = {x · y} ⊆ U. Hence the map (x, y) → x · y is continuous. (TG2 ) We show that the map I : x → x−1 is continuous. Let x ∈ G and U any neighborhood of x−1 in (G, τ1 ). Now V = {x} ∈ τ , and I(V ) = V −1 = {x−1 } ⊆ U. Thus I = x → x−1 is contains on G. Hence (G, ·, τI ) is a TG. Lemma A.8.7. Any group G = (G, ·) with the indiscrete topology τD is a TG. Proof. (TG1 ) We show that the map (x, y) → x · y is continuous. Let x, y ∈ G. Since (G, τD ) is an indiscrete space, G is the only non-empty open set in G and so U = G is the only τD -neighborhood of G. Take V = G and W = G as τD -neighborhoods of x and y respectively. Then V · W = G · G = G = U. Hence (x, y) → x · y is continuous. (TG2 ) We show that I = x → x−1 is continuous. Let x ∈ G. Then U = G is the only τD -open neighborhood of x−1 . Take V = G as τD neighborhood of x. Then I(V ) = V −1 = G−1 = G = U. So I : x → x−1 is continuous.


Thus (G, ·, τD ) is a TG.

305

Theorem A.8.8. Let G = (G, ·, τ ) be a TG. Then, for any fixed a ∈ G, the maps La : x → a · x and Ra : x → x · a are homeomorphisms of G to G. Proof. We prove the theorem for La . La is 1-1: Let x, y ∈ G with La (x) = La (y). Then a·x

= a · y =⇒ a−1 · (a · x) = a−1 · (a · y) =⇒ e · x = e · y =⇒ x = y.

La is onto: Let y ∈ G (domain) and La (a−1 · y) = a · (a−1 · y) = e · y = y. This shows that La is onto. La is continuous: Let xo ∈ G. By (TG1 ), the map (a, x) → a · x is continuous at xo or, equivalently, La : x → ax is continuous at xo . (La )−1 is continuous: Now La (x) = a · x = z (say), and so −1 L−1 · z = La−1 (z). a (z) = x = a −1 : Thus L−1 a = La−1 . By above part, La−1 is continuous. Hence (La ) G → G is continuous. Thus La : x → a · x is a homeomorphism. Similarly Ra : x → x · a is a homeomorphism.

Corollary A.8.9. Let G = (G, ·, τ ) be a TG and U an open set in G. Then, for any a ∈ G, a · U and U · a are also open. Moreover, if A ⊆ G, A · U and U · A are open. Proof. Since the mapping La : x → ax is homeomorphism, it is, in particular, an open map. Hence La (U) or a · U is open. Similarly Ra : x → x · a is open and so U · a is an open set. Next, if A ⊆ G, A·U = ∪x∈A a·U which is open; also U ·A = ∪x∈A U ·a is open. Lemma A.8.10. Let G = (G, ·, τ ) be a TG. Then: (i) The map x → x−1 is a homeomorphism. (ii) If a, b ∈ G, the map x → a · x · b is a homeomorphism. (iii) If a ∈ G, the map x → x · a · x−1 is continuous. (iv) If Z is a base of neighborhoods at e, then, for any x ∈ G, {x· U : U ∈ Z} is a base of neighborhoods at x, and so is {U · x : U ∈ U}. Theorem A.8.11. Let G = (G, ·, τ ) be a TG with identity e and let U be a base of neighborhoods at e. Then: (i) e ∈ U for each U ∈ U.

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(ii) For each U ∈ Z, there exists some V ∈ Z such that V 2 = V · V ⊆ U. (iii) For each U ∈ Z, there exists some W ∈ Z such that W −1 ⊆ U. (iv) For each U ∈ Z and a ∈ U, there exists some V ∈ Z such that a · V ⊆ U. (v) For each U ∈ Z and x ∈ G, there exists some V ∈ Z such that x · Vx−1 ⊆ U. (vi) For each U, V ∈ Z, there exists some W ∈ Z such that W ⊆ U ∩V. (vii) ∩ U = {e}. U ∈U

Proof. (i) Since each U ∈ U is a neighborhood of e and so e ∈ U for each U ∈ U (ii) Let U ∈ U. Since the mapping (x, y) → x · y is continuous at (e, e) and (e, e) → e · e = e, there exist neighborhoods V1 and V2 of e such that V1 · V2 ⊆ U. Put V = V1 ∩ V2 . Then V is a neighborhood of e, V ⊆ V1 and V ⊆ V2 . Hence V 2 = V · V ⊆ V1 ⊆ U. (iii) Let U ∈ U. Choose an open neighborhood W1 of e with W ⊆ U. Since x → x−1 is a homeomorphisms, W1−1 is an open neighborhood of e−1 = e. Let W = W1 ∩ W1−1 . Then W ∈ U and W −1 = W ⊆ U. (iv) Let U ∈ U and a ∈ U. Since the map (x, y) → x · y is continuous at (a, e) and (a, e) → a · e = a, ∃ a neighborhood V of e in X such that a · V ⊆ U. The remaining proofs are omitted. Theorem A.8.12. [Ber74, p.17] Let G = (G, ·, τ ) be a topological group with a base U of neighborhoods of e in G. Then the following are equivalent. (a) G is Hausdorff. (b) {e} is a closed subset of G. (c) ∩ U = {e}. U ∈U

Theorem A.8.13. (a) Let G be a TG. Then, for any neighborhood U of e in G, there exists a neighborhood V of 0 such that V ⊆ U. Consequently, every topological group is regular. (2) Every Hausdorff TG is completely regular. Definition. [Ber74, p. 26] A metric d on a group (G, .) is called:


307

(i) left invariant if d(a · x, a · y) = d(x, y) for all x, y, a ∈ G. (ii) right invariant if d(x · a, y · a) = d(x, y) for all x, y, a ∈ G. Metrization Theorem A.8.14. (Birkhoff-Kakutani ) ([Ber74, p. 28], [Hus66, p. 49]) (a) Let G = (G, ·, τ ) be a TG with identity element e. Then G is metrizable iff it is Hausdorff and has a countable base of τ -neighborhoods of e. (b) A mertizable TG admits a left invariant (or right invariant) compatible metric. Theorem A.8.15. (a) Every topological group (G, τ ) is a uniform space. (b) Every topological vector space (E, τ ) is a uniform space. Proof. (a) Let B be any base of τ neighborhoods of the identity e in G. For each V ∈ B, let AV = {(x, y) ∈ G × G : xy −1 ∈ V }. Then H = {AV : V ∈ B} is a uniformity on G, as follows. (u1 ) △(G) = {(x, x) : x ∈ G} ⊆ AV for all AV ∈ H: [Let AV ∈ H with V ∈ B. For any x ∈ G, xx−1 = e ∈ V and so (x, x) ∈ AV . Hence ∆ ⊆ AV for all AV ∈ H.] (u2 ) AV ∈ H =⇒ (AV )−1 ∈ H: [Let AV ∈ H with V ∈ B. We note that (AV )−1 = AV −1 : (AV )−1 = {(x, y) : (y, x) ∈ AV } = {(x, y) : yx−1 ∈ V } = {(x, y) : xy −1 = (yx−1 )−1 ∈ V −1 } = AV −1 . Hence, if AV ∈ H, then (AV )−1 = AV −1 ∈ H since V −1 ∈ B.] (u3 ) If AV ∈ H, ∃ some AW ∈ H such that (AW )2 ⊆ AV : [Let AV ∈ H with V ∈ B. Choose W ∈ B such that W W ⊆ V . We note that (AW )2 ⊆ AW W , as follows (AW )2 = AW ◦ AW = {(x, y) : ∃ some z ∈ G such that (x, z) ∈ AW and (z, y) ∈ AW } = {(x, y) : ∃ some z ∈ G such that xz −1 ∈ W and zy −1 ∈ W } ⊆ {(x, y) : ∃ some z ∈ G such that (xz −1 )(zy −1 ) ∈ W W }

308


= {(x, y) : {(x, y) : xy −1 ∈ W W } = AW W . Hence A2W ⊆ AW W ⊆ AV .] (u4 ) If AV , AW ∈ H, ∃ some AU ∈ H such that AU ⊆ AV ∩ AW : [ Let AV , AW ∈ H with V, W ∈ B Choose U ∈ B such that U ⊆ V ∩ W . We note that AV ∩ AW = AV ∩W , as follows: AV ∩ AW = {(x, y) : xy −1 ∈ V } ∩ {(x, y) : xy −1 ∈ W } = {(x, y) : xy −1 ∈ V ∩ W } = AV ∩W . Hence AU ⊆ AV ∩W ⊆ AV ∩ AW .] (u5 ) If AV ∈ H and AV ⊆ AW , then AW ∈ H: [Let AV ∈ H and AV ⊆ AW with V, W ∈ B. We note that AV ⊆ AW implies V ⊆ W , as follows. Let x ∈ V . Then xe−1 = xe = x ∈ V , and so (x, e) ∈ AV . Since AV ⊆ AW , (x, e) ∈ AW . Hence xe−1 ∈ W or x ∈ W . So V ⊆ W . Since W ∈ B, we have AW ∈ H.] (b) This follows as in (a). In fact, let (E, τ ) be a TVS, and let W be a base of neighborhoods of 0 in X. In this case, for each V ∈ W, let AV = {(x, y) ∈ E × E : x − y ∈ V }. Then H = {AV : V ∈ W} is a uniformity on E, as follows: (u1 ) ∆ = {(x, x) : x ∈ E} ⊆ AV for all AV ∈ H (since, for any x ∈ E, x − x = 0 ∈ V and so (x, x) ∈ AV ). (u2 ) For any balanced V ∈ W, A−1 = {(x, y) : (y, x) ∈ AV } = {(x, y) : y − x ∈ V } V = {(x, y) : x − y ∈ −V } = A−V . Hence, if AV ∈ H , then A−1 V = A−V ∈ H since −V ∈ W.] (u3 ). Let AV ∈ H. Choose a balanced W ∈ W such that W +W ⊆ V . Then A2W = AW ◦ AW = {(x, y) : ∃ some z ∈ E such that (x, z) ∈ AW and (z, y) ∈ AW }, = {(x, y) : ∃ some z ∈ E such that x − z ∈ W and z − y ∈ W } ⊆ {(x, y) : x − y = (x − z) + (z − y) ∈ W + W } = AW +W . Hence A2W ⊆ AW +W ⊆ AV .] (u4 ). Let AV , AW ∈ H. Choose G ∈ W such that G ⊆ V ∩ W . Then AV ∩ AW = {(x, y) : x − y ∈ V } ∩ {(x, y) : x − y ∈ W } = {(x, y) : x − y ∈ V ∩ W } = AV ∩W .


309

Hence AG ⊆ AV ∩W ⊆ AV ∩ AW .] (u5 ).Let AV ∈ H and AV ⊆ AW . This implies V ⊆ W , as follows. [Let x ∈ V . Then x − 0 = x ∈ V , and so (x, 0) ∈ AV . Since AV ⊆ AW , (x, 0) ∈ AW . Hence x − 0 ∈ W or x ∈ W . So V ⊆ W .] Since W ∈ W, we have AW ∈ H.]

310


9. Notes and Comments Section A.1. This contains some basic definitions and results from General Topology. A full discussion of the results can be found in standard text books. See, for example, Kelley [Kel55], Gillman and Jerison [GJ60], Dugundji [Dug66], Willard [Will70] and Husain [Hus77]. Section A.2. This is a collection of basic definitions and results from topological vector spaces. See, for example, Dunford-Schwartz [DS58], Edwards [Edw65], Horvath [Hor66], Husain [Hus65], Kelley and Namioka [KN63], K˝othe [Ko69], A. Robertson & W. Robertson [RR64], Rudin [Rud91], and Schaefer [Scha71]. Section A.3. Here some linear topologies on CL(E, F ), known as the G-topologies are defined. The most important cases of the G-topology are: the topology tu of uniform convergence on bounded sets, the topology tpc of precompact convergence, the topology tc of compact convergence and the topology tp of pointwise convergence. This section also includes some fundamental theorems of Functional Analysis such as the principle of uniform boundedness, Banach-Steinhaus , open mapping theorem, and closed graph theorem. All these are given in the non-locally convex setting. Section A.4. The notion of a shrinkable neighborhood of 0 in a TVS E is presented. It had first appeared in the Ph.D. thesis of R. T. Ives in 1957 (see [Kl60a]), and further studied by Klee ([Kl60a, Kl60b]). Important facts about shrinkable neighborhoods included in this section are: (A) Every Hausdorff TVS E has a base of shrinkable neighborhoods of 0. (B) If W is a shrinkable neighborhood of 0 in a TVS E, then its Minkowski functional ρW is continuous and positively homogeneous. The proofs of these important results, due to Klee [Kl60a], may not be found in the text books and therefore we have included them here. Section A.5. This includes the notions of non-Archimedean valued fields, non-Archimedean TVSs and locally F-convex TVSs. Most of these results are collected from the monograph of Prolla [Pro82] (see also [Pro77, NBB71, vR78]). We mention that an NA TVS over F or equivalently, a locally Fconvex TVS (for non-trivially valued field F) may be considered as an analogue of the usual locally convex TVS over K.


311

Section A.6. This section contains basic definitions and results from topological algebras and modules. The proofs of these results except for a few selected theorems are omitted; a full discussion of the results can be found in standard text books. See, for example, [Mal86, Mic52, Hus83, Dal00, Fra05, Zel71, Zel73]. Section A.7. Some measure theoretic terminology is considered. For a full discussion, see Halmos [Hal50], Dunford-Schwartz [DS58], Royden [Roy88]. We also include some material on the classes of tight, τ -additive and σ-additive measures on a topological spaces X. The detail may be found in the survey papers of LeCam [Lec57], Varadarajan [Var65] and Wheeler [Whe83]. Section A.8. Uniform spaces are the carriers for the notions of uniform convergence, uniform continuity, equicontinuity and like this. These notions are easily defined in metric spaces; the important quality of metric spaces for this purpose being that the distance is a notion which can be applied uniformly to pair of points without regard to their location. This quality is not possessed by topological spaces, where the neighborhoods of a point depend on the location of the point. Therefore uniform spaces will require somewhat more structure than a topology provides. The general notion of a general uniform space was introduced by A. Weil in 193; another approach is due to W.J. Tuckey (1940). See also J.R. Isbell [Isb64]. For detail of the definitions and results of this section, see [Edw65, Will70]. In this section, we have also introduced here the notion of a ”topological group” which is more general than the notion of a ”topological vector space. It is shown that every topological group is a uniform spaces. For detail see, [Hus66, Ber74].

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APPENDIX B

List of Symbols A or cl(A), the closure of a subset A of a topological space, A.1, 1.1 A0 or int A, the interior of a subset A of a topological space, A.1, 1.1 A(x) = {f (x) : f ∈ A}, A.3, 4.1 B(X), the algebra of all bounded functions ϕ : X → K ( = R or C), 1.1 Bf (X) = {ϕ ∈ B(X) : ϕ has finite support}, 1.1 Bσ (X) = {ϕ ∈ B(X) : ϕ has σ-compact support}, 1.1 Bσo (X) = {ϕ ∈ B(X) : ϕ has countable support}, 1.1 Bo (X) = {ϕ ∈ B(X) : ϕ vanishes at infinity}, 1.1 Boo (X) = {ϕ ∈ B(X) : ϕ has compact support}, 1.1 B(X)⊗E = {ϕ⊗a : ϕ ∈ B(X), a ∈ E}, where (ϕ⊗a)(x) = ϕ(x)a, x ∈ X, 1.1 Ba(X), the σ-algebra of all Baire subsets of X,, A.7 Bo(X), the σ-algebra of all Borel subsets of X, A.7 β, the strict topology on Cb (X, E), 1.1 βX, the Stone-Cech compactification of X, A.1, 1.1 C, the field of complex numbers A.2 CMr (A, Y ), the vector apace of all continuous linear right multipliers from A to Y . C(X), the algebra of all continuous functions ϕ : X → K, 1.1 Cb (X) = {ϕ ∈ C(X) : ϕ is bounded on X}, 1.1 Co (X) = {ϕ ∈ C(X) : ϕ vanishes at infinity}, 1.1 Coo (X) = {ϕ ∈ C(X) : ϕ has compact support}, 1.1 CVb (X) = {f ∈ C(X) : vf is bounded for each v ∈ V }, 1.2 CVo (X) = {f ∈ C(X) : vf vanishes at infinity for each v ∈ V }, 1.2 C(X, E), the vector space of all continuous functions f : X → E, 1.1 Cb (X, E) = {f ∈ C(X, E) : f is bounded on X}, 1.1 Co (X, E) = {f ∈ C(X, E) : f vanishes at infinity} Coo (X, E) = {f ∈ C(X, E) : f has compact support} Crc (X, E) = {f ∈ C(X, E) : f (X) is relatively compact in E} Cpc (X, E) = {f ∈ C(X, E) : f (X) is precompact in E}, 4.2 CVb (X, E) = {f ∈ C(X, E) : vf is bounded on X for each v ∈ V }, 1.2

329

330

B. LIST OF SYMBOLS

CVpc (X, E) = {f ∈ C(X, E) : vf (X) is precompact in E for all v ∈ V }, 4.2 CVo (X, E) = {f ∈ C(X, E) : vf vanishes at infinity for each v ∈ V }, 4.2 C(X) ⊗ E = {ϕ ⊗ a : ϕ ∈ C(X), a ∈ E}, 1.1 Cφ , the composition operator on CVo (X, E), Cφ (f ) = f ◦ φ with φ : X → X, 10.2 c(X), the collection of all closed subsets of X, A.7 cz(X), the collection of all cozero subsets of X, A.7 E ′ , the algebraic dual of a vector space E, A.2 E ∗ , the topological (or continuous) dual of a topological vector space E, A.2 F, a field, A.4 F (X, E), the vector space of all functions X → E, 1.1, 1.2, 9.1, 9.2 F Vo(X, E), the vector space of all functions f : X → E with vf vanishes at infinity for each v ∈ V, 1.2 Fσ , countable union of closed sets, A.1 f ≺ U means that there exists a Z ∈ z(X) such that Z ⊆ U and supp(f ) ⊆ Z, 7.1, 7.2 kf kW = ||ρW ◦ f || = sup|ρW (f (x))|, 8.1, 8.2 x∈X

Gδ , countable intersection of open sets, A.1 G(µ, W, ε) = {f ∈ Cb (X, E) : µ({x : f (x) ∈ / W }) ≤ ε}, 7.1, 7.2 γ, the weak topology on (Cb (X, E), β), 7.1 γV , the weak topology on (CVo (X, E), ωV ), 7.2 HomA (A, Y ), the vector space of all continuous left A-module homomorRphisms of A into Y, 10.3 ϕdµ, the integral of ϕ ∈ Cb (X) with respect to a scalar measure µ : x RX → K, 1.7, 8., 8.2 dmf, the integral of f ∈ Cb (X, E) with respect to a vector measure x m : X → E ∗ , 8.1, 8.2 k, the compact-open topology on C(X, E) or Cb (X, E), 1.1 K, the field of scalars, always either the field R or the complex field C, 1.1, A.1, A.2 k(X), the collection of all compact subsets of X, A.7 K + (X), the set of all non-negative constant functions on X, 1.2 χA , the characteristic function of a subset A, 1.1, 1.2, A.1 χc (X) = {λχK : λ > 0 and K ⊆ X, K compact}, 1.2 χf (X) = {λχF : λ > 0 and FP ⊆ X, F finite}, 1.2 p ℓp = ℓp (N) = {x = (xn ) ⊆ K : ∞ n=1 |xn | < ∞}, 0 < p < ∞, A.2 Rb Lp = Lp [a, b] = {f : [a, b] → K : |f (t)|p dt < ∞}, 0 < p < ∞, A.2 a

B. LIST OF SYMBOLS

331

M(A), the set of all maximal ideals of codimension 1 in A,A.6, 5.2 Mc (A)), the set of all maximal closed ideals of codimension 1 in A,A.6, 5.2 Mθ , the multiplication operator from CVo (X, E) into F (X, E) defined by Mθ (f ) = θf with θ : X → C, 9.1, 9.2 Mϕ , the multiplication operator from CVo (X, E) into F (X, E) defined by Mϕ (f ) = ϕf with ϕ : X → E, 9.1, 9.2 Md (A), the algebra of all double multipliers (S, T ) on an algebra A, 10.1 Mσ (Ba(X)), the vector space of all bounded real-valued σ-additive measures on Ba(X), A.7, 8.1 Mτ (Bo(X)), the vector space of all bounded real-valued τ -additive measures on Bo(X), A.7, 8.1 Mt (Bo(X)), the vector space of all bounded real-valued tight measures on Bo(X), A.7, 8.1 Mσ (Ba(X), E ∗ ), the vector space of all bounded E ∗ -valued σ-additive measures on Ba(X), 8.1, 8.2 Mτ (Bo(X), E ∗ ), the vector space of all bounded E ∗ -valued τ -additive measures on Bo(X), 8.1, 8.2 Mt (Bo(X), E ∗ ), the vector space of all bounded E ∗ -valued tight measures on Bo(X), 8.1, 8.2 M(µ, v, W, ε) = {f ∈ CVo (X, E) : µ∗ ({x ∈ X : vf (x) ∈ / W }) ≤ ε}, 7.2 M(D, G) = {x ∈ Y : D.x ⊆ G}, 10.2 N(ϕ, W ) = {f ∈ Cb (X, E) : (ϕf )(X) ⊆ W }, 1.1 N(A, W ) := N(χA , W ) = {f ∈ Cb (X, E) : f (A) ⊆ W }, 1.1 p, the pointwise topology on C(X, E), 1.1 pf , the seminorm on E defined by pf (x) = |f (x)|, x ∈ E, A.2 qx , the seminorm on E ∗ defined by qx (f ) = |f (x)|, f ∈ E ∗ , A.2 R, the field of real numbers, A.2 ρA , the Minkowski functional of A ⊆ E defined by ρA (x) = inf{λ > 0 : x ∈ λA}, A.2 ϕQ⊗ a, the function, 1.1 Xα , the Cartesian product of a family {Xα : α ∈ I} of topological α∈I

spaces with the product topology, A.1 σ,the σ-compact-open topology on Cb (X, E), 1.1 σo , the countable-open topology on Cb (X, E), 1.1 So+ (X), the set of all non-negative upper semi-continuous functions on X which vanish at infinity, 1.1 S(X, E), a vector subspace of F (X, E), 9.1, 9.1 SM(X), the collection of all submeasures µ : cz(X) → [0, ∞], 7.1, 7.2

332

B. LIST OF SYMBOLS

supp(f ) = closure of the set {x ∈ X : f (x) 6= 0}, where f : X → E or K, A.1, 1.1 T A, topological algebra, A.6 T V S, topological vector space, 1.1, A.2 u, the uniform topology on Cb (X, E), 1.1 U(D, W ) = {T ∈ CL(E, F ) : T (D) ⊆ W }, A.3, A.6, 4.2, 9.1, 10.1 U(D, G) = {T ∈ HomA (A, Y ) : T (D) ⊆ G}, 10.3 Ud (D, W ) = {(S, T ) ∈ Md (A) : S(D) ⊆ W and T (D) ⊆ W }, 10.1, V > 0, a Nachbin family such that, for each x ∈ X, there is a v ∈ V with v(x) 6= 0, 1.2 V Mt (Bo(X), E ∗ ) = {vm : v ∈ V, m ∈ Mt (Bo(X), E ∗ )}, 8.2 w(E, E ∗) or σ(E, E ∗ ), the weak topology on a TVS E, A.2 w(E ∗, E) or σ(E ∗ , E), the weak∗ -topology on the dual space E ∗ , A.2 ω, the first uncountable ordinal, 1.1, 6.1, 6.2 ωV , the weighted topology on CVb (X, E) or CVo (X, E), 1.2 W = W E , a base of neighbourhoods of 0 in a topological vector space E, A.2, A.3 Wπ,ϕ , the weighted composition operator on CVo (X, E), Wπ,ϕ (f ) = π · f ◦ ϕ with ϕ : X → X and π : X → L(E), 9.2 z(ϕ) = ϕ−1 (0) = {x ∈ X : ϕ(x) = 0}, a zero set of a function ϕ : X → R, A.1, A.7 z(X), the collection of all zero sets in X, A.7

APPENDIX C

Index absorbing set, A.2 algebra, A.6 Banach algebra, A.6, 10.1 F - (complete metrizable), A.6, 10.1, 10.2 faithful (or without order), A.6, 10.1, 10.2 hypo-topological, A.6, 9.2 locally bounded, A.6, 10.2 locally C ∗ -, A.6, 1A.1 locally convex, A.6 locally idempotent, A.6, 9.1, 10.2 locally m-convex, A.6 Q-, A.6, 5.1, 5.2 semisimple, simple, A.6, 5.1, 5.2 topological, A.6, 5.1, 5.2, 9.1, 9.2, 10.1, 10.2, 10.3 algebraic dual, 1.2 approximate identity (left, right, two-sided), A.6, 10.1, 10.2 uniformly bounded, A.6 approximation property, A.2, 4.1, 4.3, 4.5 balanced set, A.2 Baire set, A.7, 8.1 base of neighbourhoods, 1.1, A.1, A.2 Borel set, A.7, 8.1 bounded function, A.2, 1.1 bounded linear mapping, A.3 bounded set, A.2 Cauchy net, A.2 characteristic function χA , A.1, 1.1 compact support, A.1, 1.1 continuous extension, A.1 continuous function, A.1, 1.1, 1.2 continuous linear mapping, A.3, convex set, A.2 333

334

C. INDEX

covering dimension, A.1, 4.1 cozero set, A.1, A.7, 8.1 dominate, 1.1 equicontinuous, A.3, 4.1 essential module, 1A.2, 10.3 finite rank operator, A.2, 4.3, 5.6 F -norm, A.2 F -seminorm, A.2 Fσ -set, F -space, A.2 fundamental TVS, 10.1, 10.3 Gδ -set, A.1 Grothendieck approximation property (GAP), 4.3 Hewitt (or real) compactification υX, 5.2 homeomorphism, A.2 homomorphism, A.6, 3.2, 5.2, 1A.1,10.3 hyperplane, 1.2 ideal (left, right, two-sided, modular), A.6, 5.1, 5.2 idempotent set, 1.6, 9.3 integral, 1.7, 8.1, 8.2 isomorphism, A.6, 3.2, 5.2, 10.1, 10.3 k-space, A.1, 1.1, 1.1, 3.1 kR -space, A.1, 3.2 locally convex TVS, A.2 locally equicontinuous, 5.2 maximal ideal space, A.6, 5.2 measure, A.7, 7.1, 8.1 Baire, A.7, 7.1, 8.1 Borel, A.7, 8.1 E ∗ -valued, 8.1, 8.2 σ-additive, A.7, 8.1 Σ-regular, A.7 τ -addivite, A.7, 8.1 tight, A.7, 8.1, 8.2 measure compact, A.7, 8.1 Minkowski functional, A.2, A.3, 6.1 multiplicative linear functional, A.6, 5.2 multiplier (left, right, double), 10.1 Nachbin family, 1.2, 3.3, 4.2, 7.2, 9.1, 9.2 non-Archimedean field, A.4, 4.6

C. INDEX

non-Archimedean TVS, A.4 operator, 9.1, 9.2 multiplication, 9.1, composition, weighted composition, 9.2 P -space, A.1 partition of unity, 4.1, 4.2 precompact set, A.1, A.2, 3.2 property V, 4.1 quasi-complete TVS, A.2, 1.1 quasi-norm, A.2 quasi-ultrabarralled, A.3, 1.1 quotient map, A.2, A.3 quotient space, A.2, A.3 radical, A.6 semicontinuous (lower, upper), A.1, 1.1 seminorm, A.2, 6.2 separable, 6.1 separation property, 4.4 sequentially complete TVS, A.2, 3.1 shrinkable neighbourhood, A.4 Stone-Cech compactification βX, A.1, 5.2, 6.2 submeasure, 7.1, 7.2 submetrizable space, 6.1 separably, 6.1 support, compact, finite, σ-compact, 1.1 support of a function, A.1, 1.1 theorem, Arzéla-Ascoli, 3.1, 3.2 Banach-Steinhaus, A.3 closed graph, A.3 Hahn-Banach, A.2 M. Krein and S. Krien, 6.1 open mapping, A.3 Riesz representation, A.7, 9.2 ˇ Stone-Cech, A.1 Stone-Weierstrass, 4.1, 4.2, 4.6 Tietze extension, A.1, topological basis (orthogonal, Schauder), A.6 topological dual, A.2 topological group, A.8

335

336

C. INDEX

topological module, A.4, 10.2, 10.3 topological space, A.1 completely regular, A.1, 1.1 hemi-compact, A.1, 1.1 locally compact, A.1, 1.1 sequentially compact, A.1, A.2 σ-compact, A.1, 1.1, 6.1 topological vector space (TVS), A.2, 1.1 admissible, A.2, 4.1 complete, quasi-complete, sequentially complete, A.2 F- (complete metrizable), A.2 fundamental, A.2, 10.1 locally bounded, A.2, 4.2, 6.1, 9.1 locally convex, A.2 metrizable, 1.1, A.2 ultrabarrelled, quasi-ultrabarrelled, A.3, 1.1 ultrabornological, A.3, 1.1, 10.1 topology, compact-open (k-), 1.1, 10.1, 10.2 countable-open (σo -) Gelfand, A.6, 5.2 pointwise (p-), 1.1 strict (β-), 1.1, 10.1, 10.2 σ-compact-open (σ-), 1.1 uniform (u-), 1.1, 10.1, 10.2 weak (w(E, E ∗)-), A.2, 7.1, 7.2 weak (γt ,- mt -), 7.1 weak (γv -, mv -), 7.2 weighted (ωV -), 1.2 totally bounded (= precompact), A.2 uniform boundedness principle, A.3, 10.1, 10.2, 10.3 uniform space, A.8, 3.1 VR -space, 3.2 vanish at infinity, 1.1, 1.2 weak approximation, 7.1, 7.2 zer0-dimensional, A.1, 4.6 zero set, A.1, 7.1, 8.1

LINEAR TOPOLOGICAL SPACES OF CONTINUOUS VECTOR

LINEAR TOPOLOGICAL SPACES OF CONTINUOUS VECTOR

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