The \(b\)-Gelfand-Phillips property for locally convex spaces (Q6617349)

This article presents an extension of the Gelfand-Phillips property for Banach spaces to Hausdorff locally convex spaces. A locally convex space \(E\) is said to be \(b\)-Gelfand-Phillips if every limited set in \(E\) which is bounded in the strong topology \(\beta(E,E')\) is precompact in this topology. Several characterizations of \(b\)-Gelfand-Phillips spaces are given, and the preservation of this property under the usual operations of locally convex spaces is analyzed. Spaces \(C(X)\) of continuous functions on a Tychonoff space \(X\) which are \(b\)-Gelfand-Phillips for different topologies are also investigated.