## Abstract

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset K of C_{k}(X) is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every k_{double-struck R}-space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space C_{k}(X) is Ascoli iff C_{k}(X) is a k_{double-struck R}-space iff X is locally compact. Moreover, C_{k}(X) endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and measure-theoretic properties of ℓ_{1}, we show that the following assertions are equivalent for a Banach space E: (i) E does not contain an isomorphic copy of ℓ_{1}, (ii) every real-valued sequentially continuous map on the unit ball B_{w} with the weak topology is continuous, (iii) B_{w} is a k_{double-struck R}-space, (iv) B_{w} is an Ascoli space. We also prove that a Fréchet lcs F does not contain an isomorphic copy of ℓ_{1} iff each closed and convex bounded subset of F is Ascoli in the weak topology. Moreover we show that a Banach space E in the weak topology is Ascoli iff E is finite-dimensional. We supplement the last result by showing that a Fréchet lcs F which is a quojection is Ascoli in the weak topology iff F is either finite-dimensional or isomorphic to double-struck K^{double-struck N}, where double-struck K ∈ {double-struck R, double-struck C}.

Original language | English |
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Pages (from-to) | 119-139 |

Number of pages | 21 |

Journal | Studia Mathematica |

Volume | 233 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 2016 |

## Keywords

- Ascoli property
- Banach space
- Chet space
- Fré
- Weak topology