Free topological vector spaces (Q526790)

Let \(X\) be a Tychonoff space. In this paper, the authors define the free topological vector space \({\mathbb V}(X)\) over \(X\) as a pair consisting of a topological vector space \({\mathbb V}(X)\) and a continuous map \(i: X \to {\mathbb V}(X)\) such that every continuous map \(f\) from \(X\) to a topological vector space \(E\) gives rise to a unique continuous linear operator \(\bar{f}: {\mathbb V}(X) \to E\) with \(f = \bar{f} \circ i\), and then they study the properties of \({\mathbb V}(X)\). It is shown that such defined \({\mathbb V}(X)\) is unique up to an isomorphism of topological vector spaces, and \(X\) is embedded into \({\mathbb V}(X)\) as a closed subspace. Among many other interesting things, the authors show that (i) \({\mathbb V}(X)\) is a \(k_\omega\)-space if and only if \(X\) is a \(k_\omega\)-space; (ii) for a \(k\)-space \(X\), \({\mathbb V}(X)\) is locally convex if and only if \(X\) is discrete and countable; (iii) \({\mathbb V}(X)\) is shown to be metrizable if and only if \(X\) is finite if and only if \({\mathbb V}(X)\) is locally compact; (iv) \({\mathbb V}(X)\) is a cosmic space if and only if \(X\) is a cosmic space if and only if the free locally convex space \(L(X)\) on X is a cosmic space; (v) \({\mathbb V}(X)\) is a barreled topological vector space if and only if \(X\) is discrete.