On free locally convex spaces

Abstract

Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. We prove that the following assertions are equivalent:
(i) every functionally bounded subset of $X$ is finite,
(ii) $L(X)$ is semi-reflexive,
(iii) $L(X)$ has the Grothendieck property,
(iv) $L(X)$ is semi-Montel.
We characterize those spaces $X$, for which $L(X)$ is $c_0$-quasibarrelled, distinguished or a $(df)$-space. If $X$ is a convergent sequence, then $L(X)$ has the Glicksberg property, but the space $L(X)$ endowed with its Mackey topology does not have the Schur property.