Free Subspaces of Free Locally Convex Spaces

Gabriyelyan, Saak S.; Morris, Sidney A.

doi:https://doi.org/10.1155/2018/2924863

Journal of Function Spaces

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Abstract Introduction Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2018 | Article ID 2924863 | https://doi.org/10.1155/2018/2924863

Free Subspaces of Free Locally Convex Spaces

Saak S. Gabriyelyan¹and Sidney A. Morris^2,3

Academic Editor: Hugo Leiva

Received24 Sept 2017

Accepted10 Dec 2017

Published24 Jan 2018

Abstract

If and are Tychonoff spaces, let and be the free locally convex space over and , respectively. For general and , the question of whether can be embedded as a topological vector subspace of is difficult. The best results in the literature are that if can be embedded as a topological vector subspace of , where , then is a countable-dimensional compact metrizable space. Further, if is a finite-dimensional compact metrizable space, then can be embedded as a topological vector subspace of . In this paper, it is proved that can be embedded in as a topological vector subspace if is a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case if It is also shown that if and denote the Cantor space and the Hilbert cube , respectively, then (i) is embedded in if and only if is a zero-dimensional metrizable compact space; (ii) is embedded in if and only if is a metrizable compact space.

1. Introduction

For a Tychonoff space , we denote by , , , and the free locally convex space, the free topological vector space, the free topological group, and the free abelian topological group over , respectively. These spaces and groups always exist and are essentially unique; see [1–5].

Let and be Tychonoff spaces. The questions of whether and can be embedded as a topological vector space in and , respectively, and whether and can be embedded as a topological group in and , respectively, have been studied for many years; see for example [6–8]. The case was studied in [7] where the following result is obtained.

Theorem 1 (see [7]). If is a finite-dimensional metric compact space, then is embedded in as a topological vector space.

The proof of this theorem in [7] is nontrivial and uses the deep and powerful Kolmogorov Superposition Theorem which answered Hilbert’s 13th Problem. In the last section of this paper, we give a simpler application of the Kolmogorov Superposition Theorem to obtain Theorem 1.

In this paper we answer similar questions for when is replaced with the Cantor space or the Hilbert cube . We determine what Tychonoff spaces are such that can be embedded as a topological vector subspace in and , respectively.

It is well-known that is universal for all zero-dimensional compact metrizable spaces and is universal for all compact metrizable spaces. So it is straightforward to prove that if is a compact metrizable space, then can be embedded as a topological vector subspace in , and if is a zero-dimensional compact metrizable space, then can be embedded as a topological vector subspace of . Of course both of these embeddings are the natural ones. But are there other Tychonoff spaces such that can be embedded as a topological vector subspace of or in a less natural way? A partial answer to this question was given in [8] where this was answered in the negative for compact spaces.

Theorem 2 (see [8]). If is a compact Hausdorff space, then (i) is embedded as a topological vector subspace of if and only if is metrizable;(ii) is embedded as a topological vector subspace of if and only if is metrizable and zero-dimensional.

In Theorem 11 of this paper we prove that the condition “compact” is unnecessary, more precisely, if is embedded in or , then must to be compact.

Related questions are as follows: let be a compact metrizable space and a subspace of . What are the conditions on under which each of the following is true: (i) can be embedded as a topological vector subspace of ; (ii) can be embedded as a topological vector subspace of ; can be embedded as a topological subgroup of ; can be embedded as a topological subgroup of . For the special case complete answers to this question are given in [7, 9].

Theorem 3. For a subspace of the following are equivalent: (i)([7]) is embedded into as a topological subgroup;(ii)([9]) is embedded into as a topological vector subspace;(iii) is locally compact. In particular, can be embedded in as a topological subgroup and can be embedded in as a topological vector subspace.

For the case of free locally convex spaces on compact metrizable spaces , our Theorem 8 gives a complete description of those subspaces of with the property that can be embedded as a topological vector subspace of . Our proof is based on significant generalizations of some results in [7].

2. The Free Locally Convex Spaces on the Cantor Space and the Hilbert Cube

Let us recall (see [1]) that the free locally convex space on a Tychonoff space is a pair consisting of a locally convex space and a continuous map such that every continuous map from to any locally convex space gives rise to a unique continuous linear operator with . The free locally convex space always exists and is essentially unique. The set forms a Hamel basis for , and the map is a topological embedding [3, 10].

It is well-known (see [3]) that the dual space of is canonically isomorphic to the space of all continuous real-valued functions on . The space endowed with the pointwise topology or with the compact-open topology is denoted by and , respectively. Denote by the space endowed with the weak topology. Then the spaces and are in duality.

Firstly we note the following useful necessary condition for the existence of an embedding for free locally convex spaces which easily follows from general facts of locally convex space theory.

Proposition 4. Let and be Tychonoff spaces. (i)If the space can be embedded into as a locally convex subspace, then also the space can be embedded into as a locally convex subspace.(ii)The space can be embedded into as a locally convex subspace if and only if is an image of under a linear continuous surjection. Consequently, if can be embedded into as a locally convex subspace, then there is a linear continuous surjection from onto .

Proof. (i) follows from Theorem of [11], and (ii) follows from Theorems and of [11] applying to and .

Note that the converse assertion in (i) of Proposition 4 holds; for example, if is barrelled, see Corollary of [11]. However, the condition on to be a barrelled space is very restrictive: the space is barrelled if and only if is discrete; see Theorem of [4].

Let and be Tychonoff spaces. To obtain sufficient conditions on and for which is embedded into , by Proposition 4, it is sufficient to find conditions on and under which the converse assertion in (i) of Proposition 4 holds. The case when is compact (or, more generally, is a -space) is the most natural and important.

Recall that a Tychonoff space is called Dieudonné complete if its topology is induced by a complete uniformity. Every Lindelöf space (in particular, every -space) and every metrizable space is Dieudonné complete. We shall use the following result of Arhangelskii.

Theorem 5 (see [12]). Let and be Dieudonné complete spaces. If a linear map is continuous, then is continuous as a map .

Recall that a map is called compact-covering if for every compact subset of there is a compact subset of such that .

Proposition 6. Let be a Dieudonné complete space and let be a -space. If is a linear continuous surjection, then is also a -space and is a quotient compact-covering map. If is additionally a compact space, then is a compact space. If is compact metrizable, then so is .

Proof. Since any -space is Dieudonné complete, the map is also continuous with respect to the compact-open topologies on both spaces by Theorem 5. As is a Fréchet space, the Open Mapping Theorem [11, Theorem ] implies that is open, and hence it is a quotient map. Therefore is also a Fréchet space, so is a -space. Since and are completely metrizable and is open, the map is compact-covering by Theorem of [13]. If, in addition, is a (metrizable) compact space, then is a (separable) Banach space, so is . Thus is also a (metrizable) compact space.

The equivalence (i)(ii) in the next theorem generalizes Lemma of [7] and has an essentially simpler proof.

Proposition 7. Let be a Dieudonné complete space and let be a -space (a compact space or a metrizable compact space). Then the following assertions are equivalent: (i) can be embedded into as a locally convex subspace.(ii) can be embedded into as a locally convex subspace.(iii)There is a linear continuous surjection of onto . If (i)–(iii) hold, the space is a -space (a compact space or a metrizable compact space, resp.).

Proof. (i)(ii) and (ii)(iii) follow from Proposition 4. Let us prove (ii)(i).
Let be an embedding of locally convex spaces. Then the dual linear map is a continuous surjection by Theorems and of [11]. By Proposition 6, is a -space (a compact space or a metrizable compact space, resp.) and is a quotient compact-covering map. Hence the dual continuous map of from the space to is an embedding of into as a locally convex subspace. Observe that since and are -spaces, and are locally convex subspaces of and , respectively, by [10, 14] (for a more general assertion, see Theorem of [15]). This observation and the fact that coincides with the restriction of to imply that the map is an embedding of into .

We can now easily deduce the following main result from Proposition 7 and Lemma of [16].

Theorem 8. Let be a subspace of a compact metrizable space . Then can be embedded into as a locally convex subspace if and only if is closed.

Proof. Assume that embeds into . Then is compact by Proposition 7, and so is closed. The converse assertion follows from Lemma of [16].

It follows immediately from Theorem 8 that is not embedded in . This fact contrasts with the facts, mentioned in Theorem 3, that can be embedded as a topological subgroup of and can be embedded as a topological vector subspace of .

It turns out that the existence of a linear continuous surjection from onto in Propositions 4 and 7 is also a sufficiently strong condition as the following easy corollary of Uspenskiĭ’s theorems [17] shows.

Theorem 9 (see [7]). Let be a linear continuous surjection. If is a metrizable compact space, then so is .

Proposition 4 and Theorem 9 immediately imply the following result.

Corollary 10. Let be a Tychonoff space and let be a compact metrizable space. If can be embedded into as a locally convex subspace, then is compact and metrizable.

Corollary 10 and Theorem 2 imply the following surprising complete descriptions of Tychonoff spaces for which is embedded in and is embedded in .

Theorem 11. For a Tychonoff space the following assertions are equivalent: (i) can be embedded into as a locally convex subspace.(ii)There is a linear continuous surjection of onto .(iii) is a compact metrizable space.

Proof. (i)(ii) follows from Proposition 4, (ii)(iii) follows from Theorem 9, and (iii)(i) follows from Theorem 2.

Theorem 12. For a Tychonoff space the following assertions are equivalent: (i) can be embedded into as a locally convex subspace.(ii)There is a linear continuous surjection of onto .(iii) is a compact zero-dimensional metrizable space.

Proof. (i)(ii) follows from Proposition 4. (ii)(i): by Theorem 9, is compact and metrizable and Proposition 7 applies.
(i)(iii): If can be embedded into , then is compact and metrizable by Corollary 10. Therefore is zero-dimensional by Theorem 2. (iii)(i) follows from Theorem 2.

Corollary 13. Let be a zero-dimensional metrizable compact space and a Tychonoff space. If can be embedded into as a locally convex subspace, then is a compact zero-dimensional metrizable space.

Proof. By Theorem 12, is embedded into and Theorem 12 applies once again.

Note that the locally compact pseudocompact space is not Dieudonné complete and its Dieudonné completion is the compact space . By [18, Example ]; the restriction continuous map is bijective. However is not open because the setis not a neighborhood of zero in .

Question 14. Do there exist a compact (nonmetrizable) space and a non-Dieudonné complete subspace of such that cannot be embedded into ? What can be said about the specific case: and ?

3. Embedding into and the Kolmogorov Superposition Theorem

Now we consider an important case when a linear continuous surjection of onto can be easily constructed using the following deep result of Kolmogorov [19] in the version of Lorentz [20, Theorem ].

Theorem 15 (Kolmogorov superposition theorem). For every there exist strictly increasing functions with values in and constants such that for every function there is a function such that

Corollary 16. Let be a finite-dimensional compact metrizable space. Then there is a linear continuous surjective map from onto .

Proof. By the Embedding Theorem [21, Theorem ], every finite-dimensional compact metrizable space is embedded into for some . Now the Tietze–Urysohn Theorem shows that to prove the corollary it is sufficient to construct a linear continuous surjective map from onto .
Define the homeomorphism by . Now, using (2) we define the operator byIt is clear that is linear, and is surjective by the Kolmogorov Theorem. To check that is continuous, fix , and . Denote by the finite family of all points in of the following form:Then, if belongs to the standard open pointwise neighborhood of zero function in , we obtainThus is continuous.

Remark 17. Levin informed the authors that a linear continuous surjective map from onto can even be chosen to be open; see [22].

We now observe the following proof of Theorem 1.

Proof. Theorem 1 follows immediately from Proposition 7 and Corollary 16.

Now we consider a noncompact case, namely, when . First we recall the following definition motivated by the Kolmogorov Superposition Theorem 15.

Definition 18 (see [23]). Let be a Tychonoff space. A family is said to be basic for if each can be written as

We shall use the following noncompact version of the Kolmogorov Superposition Theorem.

Theorem 19 (see [24]). If is a finite-dimensional locally compact separable metrizable space, then has a finite basic family.

Analogously to Corollary 16 we easily obtain the following result.

Corollary 20. Let be a finite-dimensional locally compact separable metrizable space. Then there is an such that is the image of under a continuous linear surjective map .

Proof. By Theorem 19 there exists a basic family for . Then the mapis linear and surjective. Let us check that is continuous.
Fix a standard neighborhood of zero in , where is finite and . Denote by the finite set of all points in of the form , where and . If , we obtainTherefore is continuous.

Proposition 21. Let be a disjoint union of finite-dimensional locally compact separable metrizable spaces. Then there is a continuous linear surjective map from onto .

Proof. By Corollary 20, for every , we can choose an and a continuous linear surjective map . Then the mapis continuous, linear, and surjective. It is known (see [25, Section ]) that there is a linear homeomorphism from onto . Thus the map is as desired.

Now Propositions 7 and 21 immediately imply the main result of this section.

Theorem 22. If is a disjoint union of finite-dimensional locally compact separable metrizable spaces, then can be embedded into as a locally convex subspace. In particular, for every , can be embedded into as a locally convex subspace.

We conclude by noting that Theorem 22 implies that if is the countable-dimensional locally compact separable metrizable space , then can be embedded as a locally convex subspace of .

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors thank Professors M. Levin and V. Tkachuk for their advice [22, 24] and [25], respectively. The second author acknowledges the Ben Gurion University of the Negev for hospitality during which the research for this paper was undertaken.

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Copyright © 2018 Saak S. Gabriyelyan and Sidney A. Morris. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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