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 c(0)-barrelled space E is weakly Ascoli, then E is linearly isomorphic to a dense subspace of R-Gamma for some set Gamma. Consequently, a Frechet space E is weakly Ascoli iff E = R-N for some N <= omega. If X is a mu-space and a K-R-space (for example, metrizable), then C-k(X) is weakly Ascoli iff X is discrete. If X is mu-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. (C) 2017 Elsevier B.V. All rights reserved.