## Abstract

Following Banakh and Gabriyelyan (Monatshefte Math 180:39–64, 2016), a Tychonoff space X is Ascoli if every compact subset of C _{k} (X) is equicontinuous. By the classical Ascoli theorem every k-space is Ascoli. We show that a strict (LF)-space E is Ascoli iff E is a Fréchet space or E= ϕ. We prove that the strong dual Eβ′ of a Montel strict (LF)-space E is an Ascoli space iff one of the following assertions holds: (i) E is a Fréchet–Montel space, so Eβ′ is a sequential non-Fréchet–Urysohn space, or (ii) E= ϕ. Consequently, the space D(Ω) of test functions and the space of distributions D ^{′} (Ω) are not Ascoli that strengthens results of Shirai (Proc Jpn Acad 35:31–36, 1959) and Dudley (Proc Am Math Soc 27:531–534, 1971), respectively.

Original language | English |
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Pages (from-to) | 91-99 |

Number of pages | 9 |

Journal | Monatshefte fur Mathematik |

Volume | 189 |

Issue number | 1 |

DOIs | |

State | Published - 1 May 2019 |

## Keywords

- Ascoli property
- Fréchet–Urysohn space
- Montel space
- Sequential space
- Strict (LF)-space

## ASJC Scopus subject areas

- Mathematics (all)