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 CbT(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.
|State||Published - 10 Nov 2021|
- 46A03, 46E10, 46E15, 54A20