On the Ascoli property for locally convex spaces and topological groups

We characterize Ascoli spaces by showing that a Tychonoff space $X$ is Ascoli iff the canonical map from the free locally convex space $L(X)$ over $X$ into $C_k\big(C_k(X)\big)$ is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a $c_0$-barrelled space $X$ is weakly Ascoli, then $X$ is linearly isomorphic to a dense subspace of $\mathbb{R}^\Gamma$ for some $\Gamma$. Consequently, a Fr\'{e}chet space $E$ is weakly Ascoli iff $E=\mathbb{R}^N$ for some $N\leq\omega$. If $X$ is a $\mu$-space and a $k$-space (for example, metrizable), then $C_k(X)$ is weakly Ascoli iff $X$ is discrete. We prove that the weak* dual space of a Banach space $E$ is Ascoli iff $E$ is finite-dimensional.