## Abstract

Let L(X) be the free locally convex space over a Tychonoff space X. We show that the following assertions are equivalent: (i) L(X) is ℓ_{∞}-barrelled, (ii) L(X) is ℓ_{∞}-quasibarrelled, (iii) L(X) is c_{0}-barrelled, (iv) L(X) is ℵ_{0}-quasibarrelled, and (v) X is a P-space. If X is a non-discrete metrizable space, then L(X) is c_{0}-quasibarrelled but it is neither c_{0}-barrelled nor ℓ_{∞}-quasibarrelled. We prove that L(X) is a (DF)-space iff X is a countable discrete space. We show that there is a countable Tychonoff space X such that L(X) is a quasi-(DF)-space but is not a c_{0}-quasibarrelled space. For each non-metrizable compact space K, the space L(K) is a (df)-space but is not a quasi-(DF)-space. If X is a μ-space, then L(X) has the Grothendieck property iff every compact subset of X is finite. We show that L(X) has the Dunford–Pettis property for every Tychonoff space X. If X is a sequential space and a μ-space (for example, metrizable), then L(X) has the sequential Dunford–Pettis property iff X is discrete.

Original language | American English |
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Article number | 123453 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 480 |

Issue number | 2 |

DOIs | |

State | Published - 15 Dec 2019 |

## Keywords

- (DF)-, (df)- and quasi-(DF)-spaces
- Dunford–Pettis property
- Free locally convex space
- Grothendieck property
- Sequential Dunford–Pettis property
- Weak barrelledness

## All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics