The 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 Gelfand-Phillips if every limited set in $E$ is precompact in the topology on $E$ defined by barrels. Several characterizations of Gelfand-Phillips spaces are given. The problem of preservation of the Gelfand-Phillips property by standard operations over locally convex spaces is considered. Also we explore the 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 Gelfand--Phillips property of the function space $(C(X),\tau)$ implies the Gelfand--Phillips property of $(C(X),\mathcal T)$. If additionally $X$ is metrizable, then the function space $\big(C(X),\mathcal T\big)$ is Gelfand--Phillips.