Free topological vector spaces
Saak S. Gabriyelyan, Sidney A. Morris

TL;DR
This paper investigates the properties of free topological vector spaces over Tychonoff spaces, establishing conditions under which they are $k_ ext{omega}$-spaces, locally convex, barrelled, or Baire, with implications for free locally convex spaces.
Contribution
It provides a comprehensive characterization of free topological vector spaces over Tychonoff spaces, linking their topological properties to those of the underlying space, including new results on local convexity and barrelledness.
Findings
$ ext{V}(X)$ is a $k_ ext{omega}$-space iff $X$ is $k_ ext{omega}$.
If $X$ is infinite, $ ext{V}(X)$ contains a closed subspace isomorphic to $ ext{V}( ext{N})$.
$ ext{V}(X)$ is barrelled iff $X$ is discrete.
Abstract
We define and study the free topological vector space over a Tychonoff space . We prove that is a -space if and only if is a -space. If is infinite, then contains a closed vector subspace which is topologically isomorphic to . It is proved that if is a -space, then is locally convex if and only if is discrete and countable. If is a metrizable space it is shown that: (1) has countable tightness if and only if is separable, and (2) is a -space if and only if is locally compact and separable. It is proved that is a barrelled topological vector space if and only if is discrete. This result is applied to free locally convex spaces over a Tychonoff space by showing that: (1) is…
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