## Abstract

In the main result of the paper we extend Rosenthal's characterization of Banach spaces with the Schur property by showing that for a quasi-complete locally convex space E whose separable bounded sets are metrizable the following conditions are equivalent: (1) E has the Schur property, (2) E and E _{w} have the same sequentially compact sets, where E _{w} is the space E with the weak topology, (3) E and E _{w} have the same compact sets, (4) E and E _{w} have the same countably compact sets, (5) E and E _{w} have the same pseudocompact sets, (6) E and E _{w} have the same functionally bounded sets, (7) every bounded non-precompact sequence in E has a subsequence which is equivalent to the unit basis of ℓ _{1} and (8) every bounded non-precompact sequence in E has a subsequence which is discrete and C-embedded in E _{w} .

Original language | English |
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Pages (from-to) | 363-378 |

Number of pages | 16 |

Journal | Annales Academiae Scientiarum Fennicae Mathematica |

Volume | 44 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2019 |

## Keywords

- Dunford-Pettis property
- Schur property
- Sequential Dunford-Pettis property
- Weak respecting property

## ASJC Scopus subject areas

- Mathematics (all)