TY - JOUR

T1 - Free Subspaces of Free Locally Convex Spaces

AU - Gabriyelyan, Saak S.

AU - Morris, Sidney A.

N1 - Publisher Copyright:
© 2018 Saak S. Gabriyelyan and Sidney A. Morris.

PY - 2018/1/1

Y1 - 2018/1/1

N2 - If X and Y are Tychonoff spaces, let L(X) and L(Y) be the free locally convex space over X and Y, respectively. For general X and Y, the question of whether L(X) can be embedded as a topological vector subspace of L(Y) is difficult. The best results in the literature are that if L(X) can be embedded as a topological vector subspace of L(I), where I=[0,1], then X is a countable-dimensional compact metrizable space. Further, if X is a finite-dimensional compact metrizable space, then L(X) can be embedded as a topological vector subspace of L(I). In this paper, it is proved that L(X) can be embedded in L(R) as a topological vector subspace if X is a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case if X=Rn, nN. It is also shown that if G and Q denote the Cantor space and the Hilbert cube IN, respectively, then (i) L(X) is embedded in L(G) if and only if X is a zero-dimensional metrizable compact space; (ii) L(X) is embedded in L(Q) if and only if Y is a metrizable compact space.

AB - If X and Y are Tychonoff spaces, let L(X) and L(Y) be the free locally convex space over X and Y, respectively. For general X and Y, the question of whether L(X) can be embedded as a topological vector subspace of L(Y) is difficult. The best results in the literature are that if L(X) can be embedded as a topological vector subspace of L(I), where I=[0,1], then X is a countable-dimensional compact metrizable space. Further, if X is a finite-dimensional compact metrizable space, then L(X) can be embedded as a topological vector subspace of L(I). In this paper, it is proved that L(X) can be embedded in L(R) as a topological vector subspace if X is a disjoint union of a countable number of finite-dimensional locally compact separable metrizable spaces. This is the case if X=Rn, nN. It is also shown that if G and Q denote the Cantor space and the Hilbert cube IN, respectively, then (i) L(X) is embedded in L(G) if and only if X is a zero-dimensional metrizable compact space; (ii) L(X) is embedded in L(Q) if and only if Y is a metrizable compact space.

UR - http://www.scopus.com/inward/record.url?scp=85041723596&partnerID=8YFLogxK

U2 - 10.1155/2018/2924863

DO - 10.1155/2018/2924863

M3 - Article

AN - SCOPUS:85041723596

VL - 2018

JO - Journal of Function Spaces

JF - Journal of Function Spaces

SN - 2314-8896

M1 - 2924863

ER -