  Topological properties of the free topological group and the free abelian
topological group on a space have been thoroughly studied since the 1940s. In
this paper, we study the free topological $\mathbb{R}$-vector space $V(X)$ on
$X$. We show that $V(X)$ is a quotient of the free abelian topological group on
$[-1,1]\times X$, and use this to prove topological vector space analogues of
existing results for free topological groups on pseudocompact spaces. As an
application, we show that certain families of subspaces of $V(X)$ satisfy the
so-called $\textit{algebraic colimit property}$ defined in the authors'
previous work.
