The b-Gelfand–Phillips property for locally convex spaces

Abstract

We extend the well-known Gelfand–Phillips property for Banach spaces to locally convex spaces, defining a locally convex space E to be b-Gelfand–Phillips if every limited set in E,  which is bounded in the strong topology $\beta (E,E')$ on E,  is precompact in $\beta (E,E').$ Several characterizations of b-Gelfand–Phillips spaces are given. The problem of preservation of the b-Gelfand–Phillips property by standard operations over locally convex spaces is considered. Also we explore the b-Gelfand–Phillips property in spaces C(X) of continuous functions on a Tychonoff space X. If $\tau$ and ${\mathcal{T}}$ are two locally convex topologies on C(X) such that ${\mathcal{T}}_p\subseteq \tau \subseteq {\mathcal{T}}\subseteq {\mathcal{T}}_k,$ where ${\mathcal{T}}_p$ is the topology of pointwise convergence and ${\mathcal{T}}_k$ is the compact-open topology on C(X),  then the b-Gelfand–Phillips property of the function space $(C(X),\tau )$ implies the b-Gelfand–Phillips property of $(C(X),{\mathcal{T}}).$ If additionally X is metrizable, then the function space $\big (C(X),{\mathcal{T}}\big )$ is b-Gelfand–Phillips.