2021/11/11 by Тарас Банах, Banakh, Taras, Saak Gabriyelyan +1
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2111.06487
We extend the well-known Gelfand-Phillips property for Banach spaces to locally convex spaces, defining a locally convex space E to be Gelfand-Phillips if every limited set in E is precompact in the topology on E defined by barrels. Several characterizations of Gelfand-Phillips spaces are given. The problem of preservation of the Gelfand-Phillips property by standard operations over locally convex spaces is considered. Also we explore the Gelfand-Phillips property in spaces C(X) of continuous functions on a Tychonoff space X. If τ and \mathcal T are two locally convex topologies on C(X) such that \mathcal Tp⊆ τ⊆ \mathcal T⊆ \mathcal Tk, where \mathcal Tp is the topology of pointwise convergence and \mathcal Tk is the compact-open topology on C(X), then the Gelfand--Phillips property of the function space (C(X),τ) implies the Gelfand--Phillips property of (C(X),\mathcal T). If additionally X is metrizable, then the function space (C(X),\mathcal T) is Gelfand--Phillips.