Hemicompactness and the $k_{\omega }$-property of free topological vector spaces

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 $L^{(r)}(X)$ ($0<r\le 1$), the free locally pseudoconvex space $L^{(0)}(X)$, 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 $L^{(0)}(X)$ 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 $k_{\omega }$-spaces: $L_b(X)$ and $L_p(X)$ only for finite $X$, and $L^{(r)}(X)$, $0\le r\le 1$, only for countable discrete $X$.