When is a locally convex space Eberlein-Grothendieck?

TL;DR

This paper characterizes when certain locally convex spaces, especially spaces of continuous functions, are Eberlein-Grothendieck under the weak topology, revealing conditions on the underlying spaces such as compactness and $\sigma$-compactness.

Contribution

It provides a complete characterization for when spaces like $C_k(X)$ are Eberlein-Grothendieck or linearly Eberlein-Grothendieck, including new results for various classes of locally convex spaces.

Findings

$(C_k(X), w)$ is Eberlein-Grothendieck iff $X$ is $\sigma$-compact and locally compact for first-countable $X$.

$(C_k(X), w)$ is linearly Eberlein-Grothendieck iff $X$ is compact.

The class of spaces with linearly Eberlein-Grothendieck weak topology is closed under linear continuous quotients.

Abstract

In this paper we undertake a systematic study of those locally convex spaces $E$ such that $(E, w)$ is (linearly) Eberlein-Grothendieck, where $w$ is the weak topology of $E$. Let $C_{k}(X)$ be the space of continuous real-valued functions on a Tychonoff space $X$ endowed with the compact-open topology. The main results of our paper are: (1) For a first-countable space $X$ (in particular, for a metrizable $X$) the locally convex space $(C_{k}(X), w)$ is Eberlein-Grothendieck if and only if $X$ is both $\sigma$-compact and locally compact; (2) $(C_{k}(X), w)$ is linearly Eberlein-Grothendieck if and only if $X$ is compact. We characterize $E$ such that $(E, w)$ is linearly Eberlein-Grothendieck for several other important classes of locally convex spaces $E$. Also, we show that the class of $E$ for which $(E, w)$ is linearly Eberlein-Grothendieck preserves linear continuous quotients. Various illustrating examples are provided.