Hemicompactness and the $k_ω$-property of free topological vector spaces

Abstract: For a Tychonoff space $X$ we consider the vector space $E(X)$ with basis $X$ endowed with one of the following topologies: the weak topologies $L_b(X)$ and $L_p(X)$ generated by $C_b(X)$ and $C(X)$, the free locally convex space $L(X)$, the free locally $r$-convex space $L<sup>{(r)}(X)$ ($0&lt;r\le1$), the free locally pseudoconvex space $L<sup>{(0)}(X)$, and the free topological vector space $V(X)$. We prove that $L_b(X)$ is hemicompact if and only if $X$ is finite; that for every vector topology between $L_p(X)$ and $L<sup>{(0)}(X)$ hemicompactness is equivalent to $X$ being countable with all compact subsets finite; and that $V(X)$, as well as the free topological groups $F(X)$ and $A(X)$, are hemicompact if and only if $X$ is hemicompact. In particular, $V([0,1])$ is hemicompact, while $L([0,1])$ and $L_p([0,1])$ are not. We also describe when these spaces are $k_ω$-spaces: $L_b(X)$ and $L_p(X)$ only for finite $X$, and $L<sup>{(r)}(X)$, $0\le r\le1$, only for countable discrete $X$.