Free Subspaces of Free Locally Convex Spaces

Abstract

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, n∈N. 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.