Free Subspaces of Free Locally Convex Spaces

Saak S Gabriyelyan,Sidney A Morris

doi:10.1155/2018/2924863

Saak S Gabriyelyan, Sidney A Morris

Open Access

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

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Abstract

IfXandYare Tychonoff spaces, letL(X)andL(Y)be the free locally convex space overXandY, respectively. For generalXandY, the question of whetherL(X)can be embedded as a topological vector subspace ofL(Y)is difficult. The best results in the literature are that ifL(X)can be embedded as a topological vector subspace ofL(I), whereI=[0,1], thenXis a countable-dimensional compact metrizable space. Further, ifXis a finite-dimensional compact metrizable space, thenL(X)can be embedded as a topological vector subspace ofL(I). In this paper, it is proved thatL(X)can be embedded inL(R)as a topological vector subspace ifXis a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case ifX=Rn, n∈N.It is also shown that ifGandQdenote the Cantor space and the Hilbert cubeIN, respectively, then (i)L(X)is embedded inL(G)if and only ifXis a zero-dimensional metrizable compact space; (ii)L(X)is embedded inL(Q)if and only ifYis a metrizable compact space.

Highlights

For a Tychonoff space X, we denote by L(X), V(X), F(X), and A(X) the free locally convex space, the free topological vector space, the free topological group, and the free abelian topological group over X, respectively
For the case of free locally convex spaces on compact metrizable spaces Y, our Theorem 8 gives a complete description of those subspaces X of Y with the property that L(X) can be embedded as a topological vector subspace of L(Y)
We conclude by noting that Theorem 22 implies that if X is the countable-dimensional locally compact separable metrizable space ⨆n∈NRn, L(X) can be embedded as a locally convex subspace of L(R)

Summary

Introduction

For a Tychonoff space X, we denote by L(X), V(X), F(X), and A(X) the free locally convex space, the free topological vector space, the free topological group, and the free abelian topological group over X, respectively. Related questions are as follows: let Y be a compact metrizable space and X a subspace of Y. For the case of free locally convex spaces on compact metrizable spaces Y, our Theorem 8 gives a complete description of those subspaces X of Y with the property that L(X) can be embedded as a topological vector subspace of L(Y).

Results

Conclusion