## Abstract

We characterize Ascoli spaces by showing that a Tychonoff space X is Ascoli iff the canonical map from the free locally convex space L(X) over X into C_{k}(C_{k}(X)) is an embedding of locally convex spaces. We prove that an uncountable direct sum of non-trivial locally convex spaces is not Ascoli. If a c_{0}-barrelled space E is weakly Ascoli, then E is linearly isomorphic to a dense subspace of R^{Γ} for some set Γ. Consequently, a Fréchet space E is weakly Ascoli iff E=R^{N} for some N≤ω. If X is a μ-space and a k_{R}-space (for example, metrizable), then C_{k}(X) is weakly Ascoli iff X is discrete. If X is a μ-space, then the space M_{c}(X) of all regular Borel measures on X with compact support is Ascoli in the weak^{⁎} topology iff X is finite. The weak^{⁎} dual space of a metrizable barrelled space E is Ascoli iff E is finite-dimensional.

Original language | English |
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Pages (from-to) | 517-530 |

Number of pages | 14 |

Journal | Topology and its Applications |

Volume | 230 |

DOIs | |

State | Published - 1 Oct 2017 |

## Keywords

- Banach space
- Barrelled space
- Direct sum of locally convex spaces
- Free locally convex space
- Function space
- The Ascoli property