On the Ascoli property for locally convex spaces and topological groups

الأصل · EN

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 Cₖ(Cₖ(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 X is weakly Ascoli, then X is linearly isomorphic to a dense subspace of RΓ for some Γ. Consequently, a Fréchet space E is weakly Ascoli iff E=Rⁿ for some N≤ω. If X is a μ-space and a k-space (for example, metrizable), then Cₖ(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.