Free topological vector spaces
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.