Characterisation of normed linear spaces with Mazur's intersection ...

BULL. AUSTRAL. MATH. SOC. VOL.

18 ( 1978),

46BI0

105-123.

Characterisation of normed linear spaces with Mazur's intersection property J.R. Giles, D.A. Gregory, and Brailey Sims Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak * denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established.

The question of the density of the

strongly exposed points of the ball is examined for spaces with such properties.

It was Mazur [7] who drew attention to the euclidean space property (I): every bounded closed convex set can be represented as an intersection of closed balls; and he began the investigation to determine those normed linear spaces which possess this property.

Phelps [9] continued this investigation,

characterising finite dimensional spaces with property (I). Recently, Sullivan [/2] has given a characterisation of smooth spaces with property (I). Received 10 November 1977. This paper was written while the authors were visiting scholars at the University of Washington in 1977. They wish to thank the University of Washington for its hospitality and to express their appreciation at being able to participate in the Rainwater Seminar led by Professor R.R. Phelps. They wish to acknowledge their discussions with Mark Smith and Francis Sullivan. 105

106

J . R . Giles,

D.A. Gregory,

and Brailey

Sims

We develop SulIivan's key idea, and in Theorem 2.1 characterise normed linear spaces with property (I) as those with dual ball having weak * denting points norm dense in the unit sphere. In this theorem we actually give four equivalents for property (I), two of which had been given by Phelps as necessary conditions, [9, p. 979], and two others related to SuI Ii van's approach. For a dual space the appropriate intersection property which corresponds to property (J) is property (weak * I) : every bounded weak * closed convex set can be represented as an intersection of closed balls. In Theorem 3.1 we characterise Banach spaces with property (weak * I) on the dual as those with ball having denting points norm dense in the unit sphere. Phelps proved that a normed linear space has property (I) if the weak * strongly exposed points of the dual ball are norm dense in the unit sphere, [9, p. 977], and he raised the question of the necessity of this condition for property (J) . The corresponding question for property (weak * J) concerns the norm density of the strongly exposed points of the ball in the unit sphere. We show, in particular, that when a Banach space has both property (j) and property (weak * I) on the dual, then both these density properties hold. 1.

Preliminaries

We consider r e a l normed linear spaces. Given a normed linear space X , and y € X and r > 0 , we denote by B[y; r] the closed ball {x € X : \\x-y\\ 5 r} . We denote by B{X) the closed unit ball {x € X : \\x\\ 5 1} and by S(X) the unit sphere {x € X : \\x\\ = 1} . For

x € Six) , we denote by D(x) the set / € S(X*) , we denote by D^if)

{/ € S(X*) : fix) =1} , and for

the set

{a; € S(X) : fix) = 1} .

(For

every x € S(X) , D(x) is non-empty, but for some / € S(X*) , O~1(f) may be empty.) We say that X is smooth at x € Six) , if Dix) is a i >• Dix) of Six) into single point set. The set valued mapping x — subsets of Six*) is called the duality mapping on X . The inverse duality mapping on X* is the set valued mapping f \-+ D~ if)

of D[SiX))

Normed l i n e a r s p a c e s into subsets of f

S(X) . A mapping x *—*• f

of

x

107 S(X)

into

S(X*) , where

€ D{x) , is called a support mapping on X . A slice of the ball B{X) determined by

form S[B(X), f, 6) ± {x € B(X) : f(x) > 1-6}

/ € S(X*) is a set of the for some

0 < 6 < 1 . A

slice of the ball B(X*) determined by a; € S(X) is sometimes called a weak * slice of B(X*) . We say that x t S(X) is a denting point of B(X) if, for every

e > 0 , x

diameter less than

is contained in a slice of B(X) of

e . We say that / € S{X*) is a weak

point of B(X*) if, for every

e > 0 , /

slice of B(X*) of diameter less than

e . We say that x € S{X) is a

strongly exposed point of B(X) , if there exists an for every

e > 0 , f

diameter less than

* denting

is contained in a weak * / 6 S(X*) such that,

determines a slice of B{X) containing

x

and of

e . We say that f € S(X*) is a weak * strongly

exposed point of B{X*) , if there exists an x € S(X) such that, for every

determines a slice of B(,X*) containing

e > 0 , x

diameter less than

/

and of

e . A strongly exposed point is a denting point, but

the converse is not true even in finite dimensional spaces. We have the following elementary but useful property for slices. LEMMA

1.1.

In a normed linear space X , consider x € S{X) . For

any slice determined by an E > 0 such that for all

f € S(X*) and containing x

g € S(X*) , where

slice determined by g which contains x determined by

||/-g|| < e , ihere exists a and is contained in the slice

f .

Proof. Suppose for

there exists an

x € s[B(X), /, 5) . Choose

\\f-g\\ < c we have

e < %(/(x)-l+ 1 - (6-e) ; so

x € S{B(X), g, &-£.) . Also for all y € S[B(X) , g, 6-e) fly) ^ g(y) - e > 1 - 6 j so

y

i S[B(X),

we have

f, 6) .

From the Bishop-Phelps Theorem we make an immediate deduction. COROLLARY

1.1. For a Banach space X , consider x (. S{X) . For any

slice determined by an determined by a

f € S(X*) and containing x

g € D[S{X))

which contains x

there exists a slice

and is contained in -the

slice determined by f . As we might guess, conditions for property (J) must involve some

108

J . R . Giles,

D.A. Gregory,

and Brailey

Sims

smoothness condition on the unit sphere, and the problem has been to find the right condition for a characterisation.

Both Mazur and Phelps

concentrated on conditions involving strong differentiability of the norm. The norm of a normed space (Frechet differentiable)

X

at

is said to be strongly

x i S(X) , if for a l l

llm

\o

of

X

x € S(X) , then

is strongly differentiable at

is a weak

*

and real

X,

JkiMdki x

exists and is approached uniformly for all strongly differentiable at

differentiable

y 6 S{X)

y € S(X) . If the norm is X

is smooth at x . The norm

x € S{X) , if and only if /

strongly exposed point of

B(X*)

by x .

€ D(x)

The norm of X*

is strongly differentiable at / € S(X*) , if and only if there exists an x € S(X) by

/

such that

/ € D(x)

and x

is a strongly exposed point of B[X)

[70, Theorem 1 ] .

Sul livan, instead of concentrating on the set of points of strong differentiability of the norm in given

e > 0 , consider the set

such that for some

S(X) , considered more general sets. For M (X)

consisting of points

a; € S(X)

6(e, a:) > 0 ,

p

0 0 there exists a y (. S(X) n A and f € D{y) such that / determines a s l i c e of B{X) which contains x and has diameter less than y € A belongs t o this s l i c e , \\x-y\\ < e , and so x € A . (ii)

Z . As

This proof follows similarly, applying Lemma 1.1 to X* .

In p a r t i c u l a r , we can make the following deduction. COROLLARY 1.2. (i) For a normed linear space X , if for every e > 0 , D[M (X)) is norm dense in S(X*) , then fl M (X) contains the e e>0 e denting points of B{X) . (ii) D~^~[M

e

For a normed linear space

X , if for every

(.X*)) is norm dense in S(X) , then

* denting points

2.

fl M (X*) e>0 E

e >0} contains

the weak

of B(X*) .

Characterisation of spaces with property ( J )

We approach our theorem by characterising points in M (X)

sets.

Lemma 2.1 and the corresponding Lemma 3.1 generalise results given initially by Smulian, for points of strong differentiability of the norm [ M , p. 61.5]. LEMMA

2.1. For a normed linear space

X , given

e > 0 , the

following statements are equivalent: (i) (ii)

x € Mt(X) ; x

determines a slice of B{X*) of diameter less than

e ;

110

J . R . Giles,

(Hi)

D.A.

for all sequences

f

Gregory,

€ S(X*) such that

lim sup \\ffnn-fj fj (iv)

for all sequences f

x

x

3

and all

€ D[x ) , n n

Choose y

f (x) •*• 1 ,

€ S(X) where x •*• x

X

fn(x)

Sims

1 - 1/n 2 , and H/,,-^11 > e - 1/n .

such that

[f -g ) [y ) > e - 1/n .

Then

-bnh-fnl*+bn)+*n[X-i»n)

II* + ~ ||

n

- 4; * J (/„-«„)

Therefore, > e - - for a l l n

l/n

(ii) ** (Hi) . If there exists a 6 > 0 such that diam S(B(JT*), x, 6) < C , then f € .?(fl(X*), a;, 5) for

n .

n

sufficiently

large. (Hi) =» (iv).

\f

xn

(x)-l| S |f

xn

[x-x) | < ||x-x || , so that n n

,

(iv) ** Ciy). For any given j/ € S{X)

and X -»• 0+ , and for any

Normed

linear

I II

spaces

x-X and

g

€ D

we have

[5; p.- 108]. But

- Nil + |xnl So there exists a

0 < 6(e, x) < 1 , and, from (iv),

Ik+A j/||+||x-X a ||-2 r

sup 00 e

is norm dense in

112

S(X)

J.R. Giles,

, but

fi M Ax) E>0

D.A. G r e g o r y ,

i s t h e s u b s e t of

and B r a i l e y

S(X)

Sims

where t h e norm i s s t r o n g l y

differentiate. The following reflexivity result is a special case of a lemma of Sullivan [72, Lemma, §3.3], but the proof is somewhat simpler. COROLLARY

2.3. If for a Banadh space

0 < e < 1 such that

D[MAX*))

X

there exists some

is norm dense in S(X**) , then

X is

reflexive. Proof.

Since

F € D{MAX*))

e < 1 , it is sufficient to show that each

is within

dense in B(X**)

e

distance of X . Since

each weak

some element of B(X)

* slice of B(X**)

containing

. But by Lemma 2.1 (ii) ,

slice of diameter less than

B{X) is weak

F

F

*

contains

is in such a weak

*

e .

In particular, using the Bishop-Phelps Theorem we can make the following deduction from Corollary 2.3. This generalises the well known result that a Banach space

X

is reflexive if the norm of X* is strongly

differentiable on S(X*) . COROLLARY 0 < e 0 (ii)

e

for every bounded set a closed ball

(Hi)

given

containing

0 < e < 1

6(e) < 0

C with

there

C which does not contain

there exists

suoh that

inf f(C) > 0

D(y) cB{f;

an x € S(X) e)

for

exists 0 ;

and a

all

y € S{X) n B{x; 6) . Proof.

(i)

=* (ii) .

Let

C be a bounded set with

inf f(C) > 0 .

Normed

There exists a Since

/ € D(M /fe(*)J

such that B

n

k > 0 such that

linear spaces

C c B[0; k] .

there exists an

113

Choose

x € A/ ., U )

.

and an

/

€ Z?(x)

||/-/ || < z/k . Consider the sequence of closed balls

H S[nex; ( « - 1 ) E ] . We show that there exists an

C c B

e i i inf f{C) .

Suppose otherwise, that for every

n

nn 0

such that

there exists an

"o x

€ C\B

. Write

x = —

y

;

then since

{x }

is bounded,

y

•*• 0 .

But

and this contradicts

(ii) •* (Hi). such that

x i. M ,T,(.X) £ IK Consider

/(u) > e . Write

•

D = B{X) n /""""(O) , and let u' = ^ u .

Then

2 f(D+u') = ^ - > 0

0 f Z> + u' . So there exists a closed ball containing containing

0 .

not containing

u € S(X)

D + u'

be

and and not

Therefore there exists a closed ball containing

D

and

u' . The proof now follows identically that of Lemma U.I

of Phelps [9, p. 979]. (Hi) •* (i) . x € S{X)

and a

Given

6(n) > 0

y € S U ) n B(x; 6) . x

€ S(X)

where

n /

x

e > 0

and

such that

In particular ->• x

0 < n < e/2 , there exists an D(y) c B ( / ; n)

we have that

n

lim sup ||/ -/ || 5 2r| < e x x n m

€ Z?(x ) . But by Lemma 2.1 (iv) this implies that

xn n> II/-4II < n , where

for all

D(x) c B ( / ; n) , and for all for all

x i M [X) .

e / ^ € Z?(A/ £ (X)) .

The localised form of one of Mazur's results [ 7 , p . 128] follows

So

114

J.R.

Giles,

D.A.

d i r e c t l y from Lemma 2.2 COROLLARY 2.5. any of the properties

Gregory,

a n dB r a i l e y

Sims

(ii).

For a named linear space X , let f € S{X*) have of Lemma 2.2. Then, for any bounded sequence {x }

such that every closed ball containing have that f[x ) •*• fix) .

a subsequence also contains

x , we

quasi-continuous We say t h a t the duality mapping x •—•• D(x) on X i s if, given / € S(X*) and 0 < e < 1 , there exists an x € Six) and a 6(e, f) > 0 such t h a t Diy) c_ B(f; e) for a l l y € S(X) n B(x; 6) . are

THEOREM 2 . 1 . For a normed linear space equivalent: (i) (ii) (Hi)

X has property the duality

X > the following

(J);

mapping on X is

quasi-continuous;

every support mapping on X maps norm dense sets in to norm dense sets in S(X*) ;

(iv)

for every

(v)

the weak

statements

e > 0 ,

D{MAx))

* denting points

is norm dense in of

B(X*)

Six)

Six*) ;

are norm dense in

S(X*) . Proof. (i) °* (ii) . I f the duality mapping i s not quasi-continuous, there exists an / € S(X*) which does not obey the property given in Lemma 2.2 (Hi). So by Lemma 2.2 (ii) there exists a bounded closed convex set C with inf fiC) > 0 and every closed b a l l containing C also contains 0 . So then C cannot be represented as an intersection of closed b a l l s . (This i s Phelps 1 Lemma U.l [9, p . 979].) (ii) "* (Hi) . I t follows directly from the definition of quasicontinuity that for any support mapping ) on X , i f A i s norm dense in S{X) , then U) i s norm dense in S{X*) . (This i s Phelps' Corollary U.2 [ 9 , p . 980].) (Hi) •* (ii) . Suppose that the duality mapping is not quasicontinuous. Then there e x i s t s an / € S{X*) and an r > 0 such t h a t , for every x € S(X) and e > 0 , there exists a y € S(x; e) , where D(y) ^ B{f; r) . So there exists a norm dense set A in S(X) and a support mapping f , where (J>(J4) n B(f\ r) = 0 .

Normed (ii)

°* (iv) .

l i n e a r spaces

If the duality mapping i s quasi-continuous, i t follows

from Lemma 2.2 (i) that

S(X*) = fl D\M (X)) . E

e>0 (iv) =* (v) . For given D[M {X))

than

e > 0 , we have by Lemma 2.1 (ii) that

is contained in the set

which are weak e .

*

D

interior to weak

So for each

e > 0 ,

By the Baire Category Theorem are precisely the weak (v) =» (iv) .

115

*

and

*

slices of

B(X*)

S(X*)

of diameter less

is norm open and norm dense in

D D e>0

is norm dense in

denting points of

Consider

e > 0

D

, the union of points in

Given

that

/

belongs to a slice determined by

Now

D(x)

S(X*) ; but these

B(X*) .

/ € S{X*) , a weak

B(X*) .

*

denting point of

0 < r\ < z , there exists an

is contained in this slice.

S{X*) .

x € S(X)

such

a; of diameter less than

Therefore

||/ -f\\ < n

n .

for any

fx € D(x) , and by Lemma 2.1 (ii) , x 6 M U ) c M (X) . So / € D[M {X)) . (iv) ** (i) .

Consider a bounded closed convex set

y € X\C . We may assume that

y = 0 .

exists a continuous linear functional / €

C

and a point

By the Separation Theorem there /

such that

inf /(c) > 0 .

How

n D[M {X)) , so by Lemma 2.2 (ii) there exists a closed ball E e>0

containing

C

A weak

which does not contain *

0 .

denting point is an extreme point.

space an extreme point is a weak

*

In a finite dimensional

denting point, so Phelps1 result [9,

p. 980] is immediate. COROLLARY

2.6.

A finite dimensional normed linear space

property (I), if and only if the set of extreme points of in

X

B(X*)

has is dense

S(X*) . From Corollary 2.3 the following result is immediate. COROLLARY

2.7.

A Banach space

X

whose dual

X*

has property (J)

is reflexive. Using the fact that points of

B(X*)

(1 D\M (X)) £ e>0

contains the weak

and that for a reflexive space

B(X*)

*

denting

is the closed

convex hull of its denting points [4, p. 2 5 ] , we can deduce the following

116

J . R .Giles,

D.A. Gregory,

and Brailey

Sims

extension of Mazur's r e s u l t given locally in Corollary 2 . 5 .

COROLLARY 2.8. For a reflexive Banach spaae with property (J)., a bounded sequence [x } converges weakly to x > if and only if every closed ball containing a subsequence also contains

x.

In his paper [9, p. 982], Phelps asked whether X having property (I) implies that the set of weak * strongly exposed points of B(X*) is norm dense in S(X*) . In the light of Theorem 2.1 this question takes the following form. PROBLEM 2 . 1 . If X has the set of weak * denting points of B{X*) norm dense in S(X*) , is the set of weak * strongly exposed points of B(X*) norm dense in S{X*) ?

If the norm of X i s strongly differentiable on a norm dense set in S(X) , then from Theorem 2.1 (Hi) we have an affirmative answer to our question. Asplund spaces satisfy this condition. So then all reflexive Banach spaces and all separable Banach spaces with property (X) satisfy this condition. This leads us to ask the further question: PROBLEM 2 . 2 . Is every Banach space with property space?

( I ) an Asplund

We point out that if X has the property that every point of D(S(X)) is a weak * denting point of B(X*) , then the norm of X is strongly differentiable on S{X) . So from Theorem 2.1 (Hi) the set D[S(X)) of weak * strongly exposed points of B{X*) i s norm dense in S(X*) , and if X i s a Banach space, then from Theorem 1.1, X i s an Asplund space. 3.

Characterisation of spaces with property (weak

*

I)

By developing Lemmas 3.1 and 3.2 for the dual space similar to Lemmas 2.1 and 2.2 we are able to establish a characterisation theorem for dual spaces with property (weak * J) similar to that given in Theorem 2.1 for spaces with property (J). LEMMA 3 . 1 . For a Banach space X , given z > 0 , the statements are eqidvalent: (i) (ii)

foliating

f e - 1/n .

B(X**) , there exists a sequence

n

Since

€ S(X)

F (/)-*• 1 , but there exists B(^)

is weak

such that

and

I l*B-£n) (/) I < l/n Then

f{x ) •*• 1 ;

such that

|| f 0 .

Suppose that

for all

| [F -F

n i. S(X)

n

there exists a sequence \\F -F || > e

a; € S(X)

but

> e - 3/n .

So lim sup Ifc^^JI 2 e • From Lemmas 2.1 and 3.1 we make the following deduction.

/ *

€ 5(A:*) dense in

118

J . R . G i l e s ,

D . A .G r e g o r y ,

a n dB r a i l e y

COROLLARY 3 . 1 . For a normed linear space

X , if

Sims

x Z M (X) , then

x € M (X**) . LEMMA 3.2. For a Banaah space statements are equivalent: (i) (ii)

(Hi)

X , given

following

x € n D~1[M {X*)) ; e eX) for every bounded set C in X* with inf x(C) > 0 , there exists a closed dual ball containing C which does not contain 0 ; given

0 < e < 1 there exists

an f € D[S(X))

6(e) > 0 such that D~ D {g) c_B(x; e) g € D[S{X)) n B(f; 6) . Proof,

x € S(X) , the

for all

Given e > 0 , D 1{M (X*)) C D[M (X*)) ; so

(i) •* (ii) .

x € fl D\M (X*)} E e>0

and a

, and the result follows from Lemma 2.2 (ii) .

(ii) =* (Hi) .

Given

e > 0 , we have from Lemma 2.2 (Hi) , that there

exists an /' € S(X*) and a 6'(e) > 0 such that D(g) cB(x; e) for all g € S(X*) n B(f; 6') . But by the Bishop-Phelps Theorem there exists an / € D[S(X)) and a 0 < 6 < &' such that B(f; 6) X*

is uniformly rotund and so the norm of

is uniformly strongly differentiable on

X

S(X) .

References [7]

Bela Bollobas, "An extension to the theorem of Bishop and Phelps", Bull. London Math. Soc. 2 (1970), l 8 l - l 8 2 .

[2]

James B. Col H e r , "The dual of a space with the Radon-Nikodym property", Pacific J. Math. 64 (1976), 103-106.

[3D

Mahlon M. Day, Normed linear spaces, 3rd ed. (Ergebnisse der Mathematik und ihrer Grenzgebiete, 21 . Springer-Verlag, Berlin, Heidelberg, New York, 1973).

[4]

J. Diestel and J . J . Uh I , J r . , "The Radon-Nikodym theorem for Banach space valued measures", Rocky Mountain J. Math. 6 (1976), 1-U6.

[5]

J.R. Giles, "On a characterisation of d i f f e r e n t i a b i l i t y of the norm of a normed l i n e a r space", J. Austral. Math. Soc. 12 (1971), 106-llU.

161 K. John and V. Z i z l e r , "A renorming of dual spaces", Israel 12 (1972), 331-336. [7]

S. Mazur, "Uber schwache Konvergenz in den Raumen Math. 4 (1933), 128-133.

J. Math.

[iP] ", Studia

Normed

linear

spaces

123

[8]

I. Namioka and R.R. Phelps, "Banach spaces which are Asplund spaces", Duke Math. J. 42 (1975), 735-750.

[9]

R.R. Phelps, "A representation theorem for bounded convex s e t s " , Proa. Amer. Math. Soc. 11 (i960), 976-983.

[10]

R.R. Phelps, "The duality between Asplund and Eadon-Nikodym spaces" (Rainwater Seminar Notes, University of Washington, S e a t t l e , 1977)•

[Jf]

V.L. Smulian, "Sur l a derivabilite de l a norme dans l'espace de Banach", C.R. (Dokl.) Aoad. Soi. VBRS 27 (19^0), 6U3-6*»8.

[72]

Francis Sullivan, Dentability, smoothability and stronger properties in Banach spaces", Indiana Math. J. 26 (1977), 5^5-553-

Department of Mathematics, University of Newcastle, Newcastle, New South Wales; Department of Mathematics, Queen's U n i v e r s i t y , Ki ngston, Ontario, Canada; Department of Mathematics, University of New England, Armidale, New South Wales.