APPROACH VECTOR SPACES Introduction In - CiteSeerX

Houston Journal of Mathematics c University of Houston

Volume , No. ,

APPROACH VECTOR SPACES R. LOWEN AND S. VERWULGEN

Abstract. In this paper we determine what properties an approach structure has to fulfil for it to concord well with a vector space structure. Not surprisingly these conditions are more subtle than those for a topology. That the conditions we impose are the right ones follows mainly from the good categorical relationship among the different categories which play an important role in this setting, namely topological vector spaces, completely regular spaces, metrizable vector spaces and of course approach vector spaces. Keywords: approach space, approach vector space, topological vector space, pseudometric vector space, norm, prenorm, balanced, absorbing Mathematical Sciences Classification: 18B99, 46A19, 46M15, 54H13

Introduction In [4] it was shown that natural approach structures exist on normed spaces and their duals, which have the weak and weak*–topologies as their topological coreflections. These approach structures allowed to deduce approximations to fundamental theorems of functional analysis but moreover they are easily seen to be natural also from the point of view of the relation with the underlying vector space structure. In this paper we prove that this is no coincidence. We isolate the precise conditions required to have approach structures concord with the algebraic operations of a vector space. Not surprisingly we are able to show that topological vector spaces fit nicely into our framework, but the conditions for the approach case are more subtle as can be seen in the third section. We also characterize approach vector structures by means of pseudometrics and prenorms, showing surprising relations to notions existing in the vast literature on topological vector spaces. Finally we prove that categorically all is as it should be, by showing that the categories which are introduced have the right topological and algebraic properties and that the right embeddings, reflections and coreflections, are present. 1

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1. Preliminaries A comprehensive account of approach spaces can be found in [8]. From there we adopt terminology and notations. Here we confine ourselves to recalling some basic concepts, giving the necessary notations and recalling a few notions from the literature on topological vector spaces and (generalized) normed spaces. We use IP as a symbol for the set IR+ ∪ {+∞}, endowed with the usual order and algebraic operations. Let X be a set and let B ⊂ IPX . Then B is called saturated if for all ϕ ∈ IPX the following holds: ∀ > 0, ∀ω < ∞ ∃ψ ∈ B : ϕ ∧ ω ≤ ψ +  ⇒ ϕ ∈ B. If we have B ⊂ C ⊂ IPX and the above is true for all ϕ ∈ C, then we say that B is saturated in C. We write hBi for the saturation of B, that is the intersection of all saturated sets containing B or equivalently the smallest saturated set containing B. For the sequel it is useful to know the transition formulas between approach systems and gauges [8]. We recall that an approach system is a family A := ((A(x)x∈X ) of saturated ideals in PX such that for each function ϕ ∈ A(x), in the first place we have ϕ(x) = 0, and in the second place we have that for all ω < ∞ and  > 0 there exists a family of functions ϕx ∈ A(x), x ∈ X fulfilling ∀z, y ∈ X : ϕ(y) ∧ ω ≤ ϕx (z) + ϕz (y) + . If G is an ideal of (extended) p(seudo)q(uasi)-metrics, then we say G is locally saturated if, whenever e is a pq-metric such that ∀x ∈ X, ∀ε > 0, ∀ω < ∞, ∃d ∈ G : e (x, .) ∧ ω ≤ d (x, .) + ε it follows that e ∈ G. Such a locally saturated ideal is called a gauge. Given an approach system A = (A(x))x∈X on a set X, the associated gauge G is given by G = {d ∈ pq M∞ (X) | ∀x ∈ X : d[x] ∈ A(x)}, and conversely given a gauge G we get the associated approach system by letting A(x) := hG[x]i, where for h ∈ IPX×X , we have put h[x] : X −→ IP : y 7→ h(x, y) and for an arbitrary H ⊂ IPX×X we have put H[x] := {y 7→ h(x, y) | h ∈ H}. Further, if X is a group and ϕ ∈ IPX then for any x ∈ X we will write ϕ x : X −→ IP : y 7→ ϕ(y − x)

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and ϕ(2) : X × X −→ IP : (x, y) 7→ ϕ(y − x). Then for C ⊂ IPX we put C x := {ϕ x | ϕ ∈ C} and C (2) = {ϕ(2) | ϕ ∈ C}. Note that hC xi = hCi x and that C (2) [x] = C x. Furthermore in the sequel, if X is a group and ϕ ∈ IPX , we say ϕ is sub– additive if ∀x, y ∈ X : ϕ(x + y) ≤ ϕ(x) + ϕ(y). If X is a vector space (we only consider real vector spaces) we call ϕ balanced if ∀x ∈ X, ∀λ ∈ [−1, 1] : ϕ(λx) ≤ ϕ(x). We call ϕ absorbing if ∀x ∈ X, ∀ > 0, ∃δ > 0, ∀λ ∈ [−δ, δ] : ϕ(λx) ≤ . This actually means that, for all x ∈ X, the map R −→ IP : λ 7→ ϕ(λx) is continuous in 0. A prenorm is a function that is sub–additive, balanced and absorbing. It will follow from several characterizations in the sequel that this terminology is justified. However, notice already that if ϕ is balanced, respectively absorbing, then for any  > 0, {ϕ ≤ } is balanced respectively absorbing, in the usual sense. Approach uniformities, or as they were renamed in [10] “uniform gauge structures” are to approach structures what uniformities are to topologies. They can be defined in a way which very clearly demonstrates the local character of approach spaces and the uniform character of uniform gauge spaces. An ideal G consisting of p-metrics (on X) is called a uniform gauge, simply if it is saturated in the set of all p-metrics. A pair (X, G) where G is a uniform gauge will be referred to as a uniform gauge space. Note that this terminology differs from the one used in [6]. If we further saturate G in the set of all functions IPX×X we obtain what is called in [6] an approach uniformity. The type of saturation used in this context, we sometimes refer to as being uniformly saturated. Uniform gauges are the (uniform) counterpart of gauges, approach uniformities are the (uniform) counterpart of approach systems. 2. Approach groups Approach groups were introduced in [5]. We will first of all make a more detailed investigation of approach groups, in order to pinpoint the right constraints

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that are needed on an approach system for it to fit nicely with a vector space structure. Definition 1. A pair (X, A), where A = (A(x))x∈X is an approach system and X is a group, is called an approach group if the following are satisfied: (AG1) for all x ∈ X we have that A(0) x = A(x), (AG2) for all ϕ ∈ A(0), for all  > 0 and for all ω < ∞ there exists ψ ∈ A(0) such that for all x, y ∈ X ϕ(x + y) ∧ ω ≤ ψ(x) + ψ(y) + , (AG3) for all ϕ ∈ A(0) we have that ϕ(0) = 0, (AG4) the map X −→ IP : x 7→ ϕ(−x) is in A(0) for all ϕ ∈ A(0). Note that there is a redundancy in the above. In particular, in the presence of (AG1)–(AG3), it is sufficient to require that for any x ∈ X, A(x) is a saturated ideal and then it is also an approach system. In the same way as in [9], it can be shown that (AG2) is equivalent to the following condition. (AG2’) For all ϕ ∈ A(0) and for all ω < ∞ there exists ψ ∈ A(0) such that for all x, y ∈ X ϕ(x + y) ∧ ω ≤ ψ(x) + ψ(y). Then, by induction, it follows that for all ϕ ∈ A(0), for all ω < ∞ and for all n ∈ IN, there exists ψ ∈ A(0), such that for all x1 , . . . , xn ∈ X n n X X ϕ( xi ) ∧ ω ≤ ψ(xi ). i=1

i=1

The condition of interlinking between different functions of the approach system, as given in (AG2), sometimes is cumbersome, and it is therefore easier to be able to work with a base for the approach system consisting of functions where the interlinking condition is trivially fulfilled, because the functions individually have good properties. The following result shows that this is possible. The proof uses exactly the same technique as in [17]. This technique, in turn, is based on the well known proof of the Urysohn theorem on the pseudometrizability of a uniformity with a countable base. Lemma 2.1. Let (X, A) be an approach group. Then there exists a base for A(0) consisting of sub–additive and symmetric functions. Proof. For ϕ0 ∈ A(0) and ω < ∞ we shall construct a sub–additive function ϕ ∈ A(0) such that ϕ0 ∧ ω ≤ ϕ. So fix ϕ0 and ω and let (ϕm )m∈IN0 be an

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increasing sequence in A(0) such that for all x, y, z ∈ X and m ∈ IN : ϕm (x + y + z) ∧

ω ≤ ϕm+1 (x) + ϕm+1 (y) + ϕm+1 (z). 2m

Define ψ := sup (ϕm ∧ m∈IN0

ω 2m−1

),

then a short calculation reveals that for all m ∈ IN0 ψ ≤ ϕm + Let n n X X ϕ(x) := inf{ ψ(xi ) | n ∈ IN0 , xi = x}. i=1

ω 2m ,

so ψ ∈ A(0).

i=1

Then ϕ ≤ ψ, so ϕ ∈ A(0) and ϕ is sub–additive by definition. We use induction to prove that for all n ∈ IN0 , the following property P (n) holds: n n X X ω ∀ (xi )i ∈ X n ∀m ∈ IN : ϕm ( ψ(xi ). xi ) ∧ m ≤ 2 i=1 i=1 ω ≤ ψ, We already have ϕ0 ∧ ω ≤ ϕ1 ∧ ω ≤ ψ and ∀m ∈ IN0 : ϕm ∧ 2ωm ≤ ϕm ∧ 2m−1 n so P (1) is valid. Fix n and suppose P (k) is valid for k < n. Let (xi )i ∈ X and m ∈ IN. We distinguish different cases. Pn Pn Pn (1) Suppose i=1 ψ(xi ) ≥ 2ωm . Then of course ϕm ( i=1 xi )∧ 2ωm ≤ i=1 ψ(xi ). Pn ω (2) Suppose i=1 ψ(xi ) < 2m and suppose there exists k ∈ {1, . . . , n − 2} P Pn k ω ω such that i=1 ψ(xi ) < 2m+1 and i=k+2 ψ(xi ) < 2m+1 . From the Pk Pk ω induction hypothesis we know that ϕm+1 ( i=1 xi ) ∧ 2m+1 ≤ i=1 ψ(xi ) Pk ω , we have and therefore, since i=1 ψ(xi ) < 2m+1 k k X X ψ(xi ). xi ) ≤ ϕm+1 ( i=1

i=1

In the same way we obtain ϕm+1 (

n X

xi ) ≤

i=k+2

Since, by definition, ϕm+1 ∧ we also have

ω 2m

n X

ψ(xi ).

i=k+2

≤ ψ and since necessarily ψ(xk+1 )
0, ω < ∞ and x ∈ X. Because hB xi = A(x), we choose ψ ∈ B such that for all y, d(x, y) ∧ ω ≤ ψ(y − x) + . This yields that B (2) is a base for G. Finally to prove that 4 ⇒ 2, let H be a base of G consisting of translation– invariant ∞p–metrics. We note that hH[x]i = A(x), in particular A(0) has a base of sub–additive functions. Furthermore A(x) = hH[x]i = hH[0] xi = hH[0]i x = A(0) x.  3. Approach vector spaces Definition 2. A pair (X, A) consisting of a vector space X and an approach system A is called an approach vector space if the following are satisfied. (AV1) (X, A) is an approach group, (AV2) for all ϕ ∈ A(0), for all  > 0 and ω < ∞, there exists ψ ∈ A(0) such that for all x ∈ X and for all λ with |λ| ≤ 1 : ϕ(λx) ∧ ω ≤ ψ(x) + , (AV3) every ϕ ∈ A(0) is absorbing. A morphism between two approach vector spaces is defined as a linear contraction and the resulting category is denoted ApVec.

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Conditions (AV2-3) may seem surprising at first since one might expect some condition on the contractivity of scalar multiplication. However, a moment’s reflection shows that this would make no sense. Multiplication of real numbers is a continuous but not a contractive function in terms of the usual Euclidean metric of IR, and hence such a definition would be useless in our context. However the characterization of topological vector spaces by means of neighborhoods of the origin brings the solution, and leads to conditions (AV2-3). If there are no conditions given on X we will in the sequel always presume that X, with or without subscripts, is a vector space. Whenever it is necessary to mention the underlying set of X explicitly, we will denote it by X. Proposition 3.1. Let A be an approach system on X. Then the following are equivalent. (1) (X, A) is an approach vector space. (2) For all x ∈ X: A(x) = A(0) x and A(0) has a base of prenorms Proof. From lemma 2.1 we already know that A(0) has a base of sub–additive functions, say B. Let ϕ ∈ B and define ϕb (x) := sup ϕ(λx). |λ|≤1

It is easy to see that ϕb is balanced and still sub–additive. In fact ϕb is the smallest balanced function that is larger than ϕ. We fix  > 0, ω < ∞ and ψ ∈ A(0) such that ∀x ∈ X and ∀λ ∈ [−1, 1], ϕ(λx) ∧ ω ≤ ψ(x) + . Then for any x we have ϕb (x) ∧ ω ≤ ψ(x) +  and therefore ϕb ∈ A(0). In particular ϕb is absorbing and thus a prenorm. Since ϕ ≤ ϕb and since B is a base for A(0) we also have that

 A(0) = {ϕb | ϕ ∈ B} . The verification of the other implication is straightforward.



Definition 3. A local prenorm system N (on X) is an ideal of prenorms that is saturated in the set of all prenorms (on X). For example, if (X, A) is an approach vector space and we define NA to be the set of all prenorms in A(0), then NA is a local prenorm system. Every local system can be obtained in this way. Indeed, let N be a local prenorm system, then we set, for all x ∈ X, AN (x) := hN xi. It is not hard to verify that (X, AN ) is an approach vector space and NAN = N . Conversely, if (X, A) is an approach vector space then proposition 3.1 yields ANA = A. The extension of this correspondence to the functorial level yields an alternative description of

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ApVec. A linear map f : X −→ Y between approach vector spaces (X, AX ) and (Y, AY ) then is a contraction if and only if ∀ϕ ∈ NAY : ϕ ◦ f ∈ NAX . Proposition 3.2. A prenorm is finite. Proof. Let ν be a prenorm and suppose there exists an x such that ν(x) = ∞. From the fact that for all z ∈ X : ν(z) ≤ 2ν( 21 z) we deduce for all n ∈ IN : ν( 21n x) = ∞. This yields a contradiction with the fact that ν is absorbing.  Definition 4. A function d : X × X −→ IP is called a vector pseudometric (on X) if there exists a prenorm ν such that d = ν (2) . Note that, moreover, then d is indeed a pseudometric. Proposition 3.3. A function d : X × X −→ IP is a vector pseudometric if and only if d is translation–invariant and the map d[0] is a prenorm. Proof. As a plain consequence of proposition 3.2, d is finite. Symmetry as well as the triangle–inequality are easily verified.  Proposition 3.4. Let (X, A) be an approach space and let G be the gauge of A. Then the following are equivalent. (1) (X, A) is an approach vector space. (2) G has a base of vector pseudometrics. Proof. 1 ⇒ 2. Let N be the local prenorm system of A. From the proof of theorem 2.2 (2 ⇒ 4) we know that N (2) , a set of vector pseudometrics, is a gauge base for G. 2 ⇒ 1. If G has a gauge base of vector pseudometrics, say H, then we know from theorem 2.2 that (X, A) is an approach group and from the proof of that theorem (4 ⇒ 1) it follows that H[0], a set of prenorms, is a base for A(0).  The gauge of an approach vector space has a base of vector pseudometrics, so the underlying structure of an approach vector space is actually a uniform approach space [8]. We even get a uniform counterpart of ApVec. Proposition 3.5. Let (X, A) be an approach space and let Γ be the approach uniformity of the approach group (theorem 2.2). Then the following are equivalent. (1) (X, A) is an approach vector space. (2) Γ is an approach uniformity with A as underlying approach system and Γ has a base of vector pseudometrics. Moreover, if (X, AX ) and (Y, AY ) are approach vector spaces with uniformities ΓX and ΓY and f : X −→ Y is linear, then the following are equivalent.

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(1) f : (X, AX ) −→ (Y, AY ) is a contraction. (2) f : (X, ΓX ) −→ (Y, ΓY ) is a uniform contraction. Proof. This is easily established making use of the proof of theorem 2.2.



An approach vector space is considered as a vector space equipped with a suitable approach structure rather than as a set with both an approach structure and a vector space structure. With this in mind, we would like the forgetful functor ApVec −→ Vec to be topological. This is a straightforward consequence of the next statement. Theorem 3.6. Let I be a class and let for all i ∈ I, (Xi , Ai ) be an approach vector space and fi : X −→ Xi be a linear map. If Ain is the initial structure for   fi X −→ (Xi , Ai ) , i∈I

then (X, Ain ) is an approach vector space which is initial for   fi . X −→ (Xi , Ai ) i∈I

Proof. It suffices to show that (X, Ain ) is an approach vector space. For each i, let Ni be the local prenorm system of Ai (0) and let n

N := {sup νij ◦ fij | νij ∈ Nij , n ∈ IN , ij ∈ I}. j=1

Then we know that N is a base for Ain (0) and for all i ∈ I and for all xi ∈ Xi the set Ni xi is a base for Ai (xi ). Considering the functions n

n

j=1

j=1

sup νij ◦ fij x : X −→ IP : y 7→ sup νij (fij (y) − fij (x)) it follows that, for all x ∈ X, we have   n Ain (x) = {sup νij ◦ fij x | νij ∈ Nij , n ∈ IN , ij ∈ I} j=1

= hN xi = hN i x = Ain (0) x. Furthermore it is not hard to prove, using the linearity of fij , that the elements of N are all prenorms. Hence (X, Ain ) is an approach vector space.  ApVec is an (epi, extremal mono)–category: if f

(X, A) −→ (X 0 , A0 )

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is a linear contraction, then the (epi, extremal mono)–factorization is given by i

f

(X, A) −→ (f (X), A0 |f (X) ) ,→ (X 0 , A0 ). From the fact that the forgetful functor ApVec −→ Vec is topological we know that any structured sink has a final structure, but the dual of theorem 3.6 is not true in TopVec (a counterexample can be found in [2]) and hence cannot be valid in ApVec, because from corollary 3.15 we know that TopVec is finally closed in ApVec. However we have the following. Theorem 3.7. Let (X, A) be an approach vector space, let X 0 be a subspace of X and let π : X −→ X/X 0 be the canonical quotient. Furthermore let Aπ be final for π : (X, A) −→ X/X 0 . Then (X/X 0 , Aπ ) is in ApVec and is final for π : (X, A) −→ X/X 0 . Proof. It is sufficient to prove that (X/X 0 , Aπ ) is an approach vector space. Let B ⊂ IPX and define π(B) = {π(ϕ) | ϕ ∈ B}, where π(ϕ)(u) = inf π(x)=u ϕ(x). Note that π(B x) = π(B) π(x) and hπ(B)i = π hBi . Also note that π(ν) is a prenorm whenever ν is a prenorm. Define Apre on X/X 0 by \ Apre (π(x)) = π (A(y)) , y−x∈Ker π

then from [7] it follows Apre = Aπ if Apre is an approach system. Since ∀x, y ∈ X we have A(y) = A(x) (y − x), we know that y − x ∈ Ker π implies π(A(x)) = π(A(y)). Thus ∀x ∈ X we have that Apre (π(x)) = π(A(x)). So Apre (π(x)) = Apre (0) π(x). If N is the local prenorm system of A, π(N ) is a base for Apre consisting of prenorms. So Apre satisfies (AV1)–(AV3), in particular it satisfies (AG1)–(AG3) and hence it is an approach system.  f

ApVec is an (extremal epi, mono)–category: if (X, A) −→ (X 0 , A0 ) is a linear contraction, then the (extremal epi, mono)–factorization is given by π

fπ

(X, A) −→ (X/Kerf, Aπ ) −→ (X 0 , A0 ). Proposition 3.8. Let I be a set, let ((Xi , Ai ))i∈I be a family of approach vector fi

spaces and let (Xi −→ qXi )i be the algebraic direct sum of (Xi )i . Then   fi (Xi , Ai ) −→ (qXi , ΣAi ) i

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is the coproduct of ((Xi , Ai ))i . Here ΣAi is defined in the following way: if Ni is the local prenorm system of Ai then for (xi )i ∈ qXi we set * + X ΣAi ((xi )i ) := {(yi )i 7→ νi (yi − xi ) | ∀i ∈ I : νi ∈ Ni } . i∈I

(An infinite sum of zero’s has to be considered zero.) Proof. Straightforward.



Definition 5. Let d be a vector pseudometric and let Ad be the associated approach structure. Then the pair (X, Ad ) is called a pseudometric vector space. A morphism between pseudometric vector spaces is a linear contraction and the category thus obtained is written as pMetVec. Let d be a vector pseudometric and let G be the gauge of Ad . Since {d} is a base for G, proposition 3.4 yields (X, Ad ) is an approach vector space. Moreover we have the following. Theorem 3.9. pMetVec is a full subcategory of ApVec. Moreover, this embedding is an extension of the embedding pMet ,→ UAp, in the sense that the following implication holds: if (X, A) is an approach vector space with the additional property that (X, A) is a pseudometric approach space, then (X, A) is a pseudometric vector space. Proof. In order to prove the second part of the proposition, let (X, A) be an approach vector space where A is a pseudometric approach structure with gauge G. This means that there exists a pseudometric d such that {d} is a base for G. We know from proposition 3.4 that this gauge has a base of vector pseudometrics, say H. Hence, d = sup H, this yields d = d(2) [0] and d[0] is balanced. What we have to show is that d[0] is absorbing. Fix x ∈ X and  > 0. Let d0 ∈ H be such that for all y ∈ X : d(0, y) ∧ ω ≤ d0 (0, y) + 2 , where ω >  is fixed. Let δ > 0 such that ∀λ ∈ [−δ, δ] : d0 (0, λx) ≤ 2 . Then we have ∀λ ∈ [−δ, δ] : d(0, λx) ≤ .  Theorem 3.10. pMetVec is initially dense in ApVec.

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Proof. An object in ApVec is in the initial hull of pMetVec if and only if its gauge has a base of vector pseudometrics. By proposition 3.4, this is the case for any object in ApVec.  In the following result part of the relation between pseudometric vector spaces and topological vector spaces is clarified. Theorem 3.11. Let X be a vector space and let T be a metrizable topology on X. Then (X, T ) is a topological vector space if and only if there there exists a vector pseudometric d such that Td = T . Proof. The construction of a vector pseudometric for a metrizable topology of a topological vector space is due to Urysohn and can be found e.g. in Schaefer [15]. We only prove the if part of the proposition. Hereto let d be a vector pseudometric on X and let (x0 , y0 ) ∈ X × X. Note that ∀x, y ∈ X d(x0 + y0 , x + y) ≤ d(x0 , x) + d(y0 , y). For a neighborhood V of x0 + y0 in Td we fix  > 0 such that {z | d(x0 + y0 , z) ≤ } ⊂ V. This yields n o n o x | d(x0 , x) ≤ + y | d(y0 , y) ≤ ⊂ V. 2 2 In order to prove the continuity of the multiplication, fix λ0 ∈ IR and x0 ∈ X. Now let V be a neighborhood of λ0 x0 in Td and let  > 0 be such that {z | d(λ0 x0 , z) ≤ } ⊂ V. Then, using the fact that d[0] is absorbing, we can find δ > 0 such that |λ − λ0 | ≤ δ ⇒ d(λ0 x0 , λx0 ) ≤

 . 2

Choose n ∈ IN0 such that |λ0 | + δ ≤ n. We claim that n  o [λ0 − δ, λ0 + δ] z | d(x0 , z) ≤ ⊂ V. 2n

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Indeed, if λ ∈ [λ0 − δ, λ0 + δ] and d(x0 , z) ≤

 2n

we have

d(λ0 x0 , λz) ≤ d(λ0 x0 , λx0 ) + d(λx0 , λz)  λ λ ≤ + d( nx0 , nz) 2 n n  ≤ + d(nx0 , nz) 2  ≤ + nd(x0 , z) 2 ≤ .  Theorem 3.12. The embedding CReg ,→ UAp generates a full embedding of TopVec into ApVec. Proof. Let (X, T ) be a topological vector space and let AT be the approach system generated by T . Then V, the neighborhood system of T , has the following properties, [15]: (1) (2) (3) (4)

V is translation invariant, ∀V ∈ V(0) ∃U ∈ V(0) : U + U ⊂ V , every V ∈ V(0) is absorbing, V(0) has a base of balanced sets.

The first two properties follow from the fact that (X, AT ) is an approach group and the last ones are equivalent with AT satisfying (AV2) and (AV3) in the definition of approach vector spaces.  From now on we will make no distinction between a topology T and the associated approach system AT . Theorem 3.13. TopVec is initially closed in ApVec. Proof. We know that CReg is initially closed in UAp, hence the result follows from theorem 3.6.  Theorem 3.14. TopVec is concretely coreflective in ApVec. If (X, A) is an approach vector space and TA is the topological coreflection of A, then we have the following. (1) (X, TA ) is a topological vector space (2) The coreflection of (X, A) is given by id

(X, TA ) −→ (X, A).

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Proof. We only show the first part of the proposition, since then the second part is a consequence. Let (X, A) be an approach vector space. By theorem 3.10 there is an initial source ((X, A) −→ (Xi , Adi ))i in pMetVec. Theorem 3.6 implies that ((X, A) −→ (Xi , Adi ))i is initial too. Since initial sources are preserved by coreflections, ((X, TA ) −→ (Xi , Tdi ))i is initial in CReg and, because CReg is initially closed in UAp, this source is initial in UAp. We know from theorem 3.11 that (Xi , Tdi ) are topological vector spaces and we conclude, by theorem 3.13 in connection with theorem 3.6 that (X, TA ) is a topological vector space.  Corollary 3.15. TopVec is finally closed in ApVec. Applying the different descriptions of approach vector spaces to topological vector spaces we obtain several equivalent descriptions of TopVec. Corollary 3.16. Let T be a topology on X with neighborhood system V and let U be the filter that is generated by the base {UV | V ∈ V(0)}, where for any V , UV := {(x, y) | y − x ∈ V }. Then the following are equivalent. (1) (X, T ) is a topological vector space. (2) V is translation invariant and there exists a set B of prenorms such that {{ν ≤ } | ν ∈ B,  > 0} is a base for VT (0). (3) U is a uniformity that has a base of vector pseudometrics and T is the underlying topology. (4) T is generated by a set of vector pseudometrics. The following diagram summarizes the main results of this paper. pMetVec 



  pMetTopVec 

/ ApVec c

/ TopVec pMet 



  pMetTop 

 / UAp c

 / CReg

The categories in the top layer are the “approach versions” of those on the bottom layer. Moreover, the dashed arrows are forgetful functors in such a way

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that the square on the right is extended to the one on the left. In particular, the horizontal embeddings are initially dense. References [1] Ad´ amek J., Herlich H. and Strecker G, Abstract and concrete categories, J. Wiley and Sons, 1990 ´ ements de math´ [2] Bourbaki N. El´ ematique, livre V, Espaces Vectoriels Topologiques, Hermann, Paris 1964 [3] Lowen E. and Lowen R. A quasitopos containing CONV and MET as full subcategories, Int. J. Math. Math. Sci. 11 (1988) pp. 417–438 [4] Lowen R. and Sioen M. Approximations in Functional Analysis, Result. Math. 37 (2000) pp. 345–372 [5] Lowen R. and Windels B. Approach groups, Rocky Mountain J. of Math. 30 (2000) pp. 1057–1074 [6] Lowen R. and Windels B. AUnif, a common supercategory of pMET and Unif, Int. J. Math. & Math. Sci. 21 (1998) pp. 1–18 [7] Lowen E., Lowen R. and Verbeeck F. Exponential objects in the construct PRAP Cah. Top. G´ eom. Diff. Cat. 38(4) (1997) pp. 259–276 [8] Lowen R. Approach Spaces: The Missing Link in the Topology–Uniformity–Metric Triad, Oxford Mathematical Monographs, Oxford University Press, 1997 [9] Lowen R. Approach spaces A common Supercategory of TOP and Met, Math. Nachr. 141 (1989) pp. 183–226 [10] Lowen R. An Ascoli theorem in approach theory, Topology Appl. to appear [11] Mac Laine S Categories for the Working Mathematician, Springer 1998 [12] Preuss G. Theory of Topological Structures: an approach to categorical topology, Mathematics and its applications, Reidel, Dortrecht, 1988 [13] Rolewicz S. Metric Linear Spaces, D. Reidel Publishing Company, 1985 [14] Rudin W. Functional Analysis, Intern. Series in Pure and Appl. Math., McGraw-Hill, 1991 [15] Schaefer H.H. Topological Vector Spaces, Graduated texts in mathematics, Springer Verlag, 1971 [16] Sioen M. Extension Theory for Approach Spaces, PhD thesis, University of Antwerp, 1997 [17] Windels B. On the metrization lemma for uniform spaces, Quaestiones Math 24 (2001) pp. 123-128. [18] Windels B. Uniform Approach theory, PhD thesis, University of Antwerp, 1997 (R. Lowen) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ANTWERP, ANTWERP 2020, BELGIUM E-mail address: [email protected] (S. Verwulgen) DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ANTWERP, ANTWERP 2020, BELGIUM E-mail address: [email protected]