The b-Gelfand–Phillips property for locally convex spaces

T. Banakh, S. Gabriyelyan

Research output: Contribution to journalArticlepeer-review

Abstract

We extend the well-known Gelfand–Phillips property for Banach spaces to locally convex spaces, defining a locally convex space E to be b-Gelfand–Phillips if every limited set in E, which is bounded in the strong topology β(E, E) on E, is precompact in β(E, E). Several characterizations of b-Gelfand–Phillips spaces are given. The problem of preservation of the b-Gelfand–Phillips property by standard operations over locally convex spaces is considered. Also we explore the b-Gelfand–Phillips property in spaces C(X) of continuous functions on a Tychonoff space X. If τ and T are two locally convex topologies on C(X) such that Tp⊆ τ⊆ T⊆ Tk, where Tp is the topology of pointwise convergence and Tk is the compact-open topology on C(X), then the b-Gelfand–Phillips property of the function space (C(X) , τ) implies the b-Gelfand–Phillips property of (C(X) , T). If additionally X is metrizable, then the function space (C(X) , T) is b-Gelfand–Phillips.

Original languageEnglish
JournalCollectanea Mathematica
DOIs
StatePublished - 6 Aug 2023

Keywords

  • Banach space
  • Function space
  • Locally convex space
  • b-Gelfand–Phillips property

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics

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