## Abstract

We show that if a locally convex space (lcs) (E, τ) is a sequentially Ascoli space in the weak topology, then τ= σ(E, E^{′}). Consequently, a complete lcs E is weakly sequentially Ascoli iff E= F^{λ} for some cardinal λ. For a Tychonoff space X, let C_{k}(X) be the space C(X) of all continuous functions on X endowed with the compact-open topology. We prove that: (1) C_{k}(X) is weakly sequentially Ascoli iff C_{k}(X) is weakly Ascoli iff it is weakly κ -Fréchet–Urysohn iff X has the property (κ) and every compact subset of X is finite; (2) C_{k}(X) is weakly Fréchet–Urysohn iff C_{k}(X) is a weakly sequential space iff it is a weakly k-space iff the space X has the property γ whose compact subsets are finite; (3) the dual space of C_{k}(X) is weak ^{∗} sequentially Ascoli iff X is finite. If C_{b}(X) is the space C(X) with the topology of uniform convergence on functionally bounded subsets of X, then: (a) C_{b}(X) is weakly sequentially Ascoli iff C_{b}(X) is weakly Ascoli iff it is weakly κ -Fréchet–Urysohn iff X has the property (κ) and every functionally bounded subset of X is finite; (2) C_{b}(X) is weakly Fréchet–Urysohn iff C_{b}(X) is a weakly sequential space iff it is a weakly k-space iff the space X has the property γ whose functionally bounded subsets are finite.

Original language | English |
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Title of host publication | Functional Analysis and Continuous Optimization - In Honour of Juan Carlos Ferrando’s 65th Birthday, 2022 |

Editors | José M. Amigó, María J. Cánovas, Marco A. López-Cerdá, Manuel López-Pellicer |

Publisher | Springer |

Pages | 123-135 |

Number of pages | 13 |

ISBN (Print) | 9783031300134 |

DOIs | |

State | Published - 1 Jan 2023 |

Event | International Meeting on Functional Analysis and Continuous Optimization, IMFACO 2022 - Elche , Spain Duration: 16 Jun 2022 → 17 Jun 2022 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 424 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Conference

Conference | International Meeting on Functional Analysis and Continuous Optimization, IMFACO 2022 |
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Country/Territory | Spain |

City | Elche |

Period | 16/06/22 → 17/06/22 |

## Keywords

- Function space
- Sequential space
- Sequentially Ascoli space
- Weak topology
- Weak topology

## ASJC Scopus subject areas

- General Mathematics

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