## Abstract

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.

Original language | English |
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Article number | 27 |

Journal | Annals of Functional Analysis |

Volume | 14 |

Issue number | 2 |

DOIs | |

State | Published - 1 Apr 2023 |

## Keywords

- Banach space
- function space
- Locally convex space
- The strong Gelfand–Phillips property

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory