The Gelfand-Phillips property for locally convex spaces

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 τ and T are two locally convex topologies on C(X) such that Tp⊆τ⊆T⊆Tk, where Tp is the topology of pointwise convergence and Tk is the compact-open topology on C(X), then the Gelfand--Phillips property of the function space (C(X),τ) implies the Gelfand--Phillips property of (C(X),T). If additionally X is metrizable, then the function space (C(X),T) is Gelfand--Phillips.