The b-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 b-Gelfand–Phillips if every limited set in E, which is bounded in the strong topology β(E,E^′) on E, is precompact in β(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 τ and T are two locally convex topologies on C(X) such that T_p⊆τ⊆T⊆T_k, where T_p is the topology of pointwise convergence and T_k is the compact-open topology on C(X), then the b-Gelfand–Phillips property of the function space (C(X),τ) implies the b-Gelfand–Phillips property of (C(X),T). If additionally X is metrizable, then the function space (C(X),T) is b-Gelfand–Phillips.