Free Subspaces of Free Locally Convex Spaces

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 (I), where I = [0, 1], then is a countabledimensional compact metrizable space. Further, if is a finite-dimensional compact metrizable space, then ( ) can be embedded as a topological vector subspace of (I). In this paper, it is proved that ( ) can be embedded in (R) 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 = R , ∈ N. It is also shown that if G and denote the Cantor space and the Hilbert cube I N , respectively, then (i) ( ) is embedded in (G) if and only if is a zero-dimensional metrizable compact space; (ii) ( ) is embedded in ( ) if and only if is a metrizable compact space.

Locally convex properties of free locally convex spaces

“…In what follows we use the following generalization of this remarkable result which is of independent interest (our detailed proof follows the Odell-Stegall idea, cf. also Theorem 5.21 of [24]).…”

“…By Theorem 4.14 of [14], the Banach space L ∞ (µ) is injective. Therefore, by Lemma 5.20 of [24], L ∞ (µ) is an injective locally convex space. In particular, the operator…”

“…Proof. Since τ and T are compatible, the spaces (E, τ ) and (E, T ) have the same (p, q)-limited subsets by Lemma 3.1(vii) of [24]. As by assumption all (p, q)-limited sets in (E, T ) are precompact (resp., sequentially precompact, relatively compact or relatively sequentially compact), the inclusion τ ⊆ T implies that so are all (p, q)-limited sets also in (E, τ ).…”

“…Proof. (i) follows from the easy fact that every (p, q)-limited set is (p, q)-E-limited, see Lemma 3.1(i) of [24], and from the definition of p-barrelled spaces.…”

“…(ii) follows from the fact that every (p, q)-(E-)limited set is (p ′ , q ′ )-(E-)limited, see Lemma 3.1(vi) of [24].…”