Locally convex spaces with the strong Gelfand-Phillips property

TL;DR

This paper introduces the strong Gelfand-Phillips property for locally convex spaces, providing characterizations and conditions under which certain function spaces possess this property, especially in relation to the topology and structure of the underlying space.

Contribution

It defines the strong Gelfand-Phillips property for locally convex spaces and characterizes it for spaces of continuous functions with various topologies, linking it to the topological structure of the underlying space.

Findings

Characterization of the strong Gelfand-Phillips property in locally convex spaces.

Conditions for $C_{ au}(X)$ to have the property based on the structure of $X$.

Identification of when $C^b_{ au}(X)$ has the property in terms of $X$ being a compact countable space.

Abstract

We introduce the strong Gelfand-Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand-Phillips property among locally convex spaces admitting a stronger Banach space topology. If $C_{\mathcal T}(X)$ is a space of continuous functions on a Tychonoff space $X$, endowed with a locally convex topology $\mathcal T$ between the pointwise topology and the compact-open topology, then: (a) the space $C_{\mathcal T}(X)$ has the strong Gelfand-Phillips property iff $X$ contains a compact countable subspace $K\subseteq X$ of finite scattered height such that for every functionally bounded set $B\subseteq X$ the complement $B\setminus K$ is finite, (b) the subspace $C^b_{\mathcal T}(X)$ of $C_{\mathcal T}(X)$ consisting of all bounded functions on $X$ has the strong Gelfand-Phillips property iff $X$ is a compact countable space of finite scattered height.