Locally convex spaces with the strong Gelfand-Phillips property

Abstract

We introduce the strong Gelfand-Phillips property for locally convex spaces and give several characterizations of this property. We characterize the strong Gelfand-Phillips property among locally convex spaces admitting a stronger Banach space topology. If CT (X) is a space of continuous functions on a Tychonoff space X, endowed with a locally convex topology T between the pointwise topology and the compact-open topology, then: (a) the space CT (X) has the strong Gelfand-Phillips property iff X contains a compact countable subspace K ⊆ X of finite scattered height such that for every functionally bounded set B ⊆ X the complement B \ K is finite, (b) the subspace C b T (X) of CT (X) consisting of all bounded functions on X has the strong Gelfand-Phillips property iff X is a compact countable space of finite scattered height.