Locally convex properties of free locally convex spaces

Let $L(X)$ be the free locally convex space over a Tychonoff space $X$. We show that the following assertions are equivalent: (i) $L(X)$ is ${\mathrm{\ell }}_{\mathrm{\infty }}$-barrelled, (ii) $L(X)$ is ${\mathrm{\ell }}_{\mathrm{\infty }}$-quasibarrelled, (iii) $L(X)$ is $c_0$-barrelled, (iv) $L(X)$ is ${\mathrm{\aleph }}_0$-quasibarrelled, and (v) $X$ is a $P$-space. If $X$ is a non-discrete metrizable space, then $L(X)$ is $c_0$-quasibarrelled but it is neither $c_0$-barrelled nor ${\mathrm{\ell }}_{\mathrm{\infty }}$-quasibarrelled. We prove that $L(X)$ is a $(DF)$-space iff $X$ is a countable discrete space. We show that there is a countable Tychonoff space $X$ such that $L(X)$ is a quasi-$(DF)$-space but is not a $c_0$-quasibarrelled space. For each non-metrizable compact space $K$, the space $L(K)$ is a $(df)$-space but is not a quasi-$(DF)$-space. If $X$ is a $\mu$-space, then $L(X)$ has the Grothendieck property iff every compact subset of $X$ is finite. We show that $L(X)$ has the Dunford–Pettis property for every Tychonoff space $X$. If $X$ is a sequential $\mu$-space (for example, metrizable), then $L(X)$ has the sequential Dunford–Pettis property iff $X$ is discrete.