## Abstract

Let X be a Tychonoff space, and let S be a directed family of functionally bounded subsets of X containing all finite subsets of X. Denote by CTS(X) the space of all continuous functions on X endowed with the topology of uniform convergence on the sets of the family S. We characterize X for which the space CTS(X) endowed with the weak topology satisfies numerous weak barrelledness conditions or (DF)-type properties, or it has a locally convex property stronger than the property of being a Mackey space. It is shown that the dual space of CTS(X) is weak^{∗} sequentially Ascoli iff X is finite. We prove also that if CTS(X) is an ℓ_{∞}-quasibarrelled space, then the strong dual of CTS(X) is a weakly sequentially Ascoli space iff X is finite.

Original language | English |
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Article number | 20 |

Journal | Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas |

Volume | 117 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2023 |

## Keywords

- Baire space
- Feral
- Function space
- Sequentially Ascoli space
- Weak barrelledness condition
- Weak topology
- Čech-complete space

## ASJC Scopus subject areas

- Analysis
- Algebra and Number Theory
- Geometry and Topology
- Computational Mathematics
- Applied Mathematics

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