Free topological vector spaces

Abstract

We define and study the free topological vector space $\mathbb{V}(X)$ over a Tychonoff space $X$. We prove that $\mathbb{V}(X)$ is a $k_\omega$-space if and only if $X$ is a $k_\omega$-space. If $X$ is infinite, then $\mathbb{V}(X)$ contains a closed vector subspace which is topologically isomorphic to $\mathbb{V}(\mathbb{N})$. It is proved that if $X$ is a $k$-space, then $\mathbb{V}(X)$ is locally convex if and only if $X$ is discrete and countable. If $X$ is a metrizable space it is shown that: (1) $\mathbb{V}(X)$ has countable tightness if and only if $X$ is separable, and (2) $\mathbb{V}(X)$ is a $k$-space if and only if $X$ is locally compact and separable. It is proved that $\mathbb{V}(X)$ is a barrelled topological vector space if and only if $X$ is discrete. This result is applied to free locally convex spaces $L(X)$ over a Tychonoff space $X$ by showing that: (1) $L(X)$ is quasibarrelled if and only if $L(X)$ is barrelled if and only if $X$ is discrete, and (2) $L(X)$ is a Baire space if and only if $X$ is finite.