Locally convex spaces with the strong Gelfand–Phillips property

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 C_T(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 C_T(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 CTb(X) of C_T(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.