On the Ascoli property for locally convex spaces and topological groups

TL;DR

This paper characterizes Ascoli spaces via embeddings of free locally convex spaces, explores conditions under which various locally convex spaces are Ascoli or weakly Ascoli, and identifies when dual spaces are Ascoli.

Contribution

It provides a new characterization of Ascoli spaces through canonical embeddings and establishes criteria for when certain locally convex spaces and duals are Ascoli or weakly Ascoli.

Findings

Uncountable direct sums of non-trivial locally convex spaces are not Ascoli.

Weakly Ascoli $c_0$-barrelled spaces are dense subspaces of $ eal^ ame$.

The dual of a Banach space is Ascoli iff the space is finite-dimensional.

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 $C_k\big(C_k(X)\big)$ 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.