On the Ascoli property for locally convex spaces and topological groups

Abstract

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 $X$ is weakly Ascoli, then $X$ is linearly isomorphic to a dense subspace of ${\mathbb{R}}^{\Gamma }$ for some $\Gamma$. Consequently, a Fréchet space $E$ is weakly Ascoli iff $E={\mathbb{R}}^N$ for some $N\le \omega$. 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.

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• Fans and their applications in General Topology, Functional Analysis and Topological Algebra
• A characterization of free locally convex spaces over metrizable spaces which have countable tightness
• The Ascoli property for function spaces and the weak topology of Banach and Fréchet spaces
• On topological properties of the weak topology of a Banach space