On the Ascoli property for locally convex spaces

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(C_k(X)) 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 c₀-barrelled space E is weakly Ascoli, then E is linearly isomorphic to a dense subspace of R^Γ for some set Γ. Consequently, a Fréchet space E is weakly Ascoli iff E=R^N for some N≤ω. If X is a μ-space and a k_R-space (for example, metrizable), then C_k(X) is weakly Ascoli iff X is discrete. If X is a μ-space, then the space M_c(X) of all regular Borel measures on X with compact support is Ascoli in the weak^⁎ topology iff X is finite. The weak^⁎ dual space of a metrizable barrelled space E is Ascoli iff E is finite-dimensional.