## 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 T

_{p}⊆ τ⊆ T⊆ T

_{k}, where T

_{p}is the topology of pointwise convergence and T

_{k}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 language | English |
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Journal | Collectanea Mathematica |

DOIs | |

State | Published - 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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