Locally convex spaces and Schur type properties

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7 Scopus citations

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 languageEnglish
Pages (from-to)363-378
Number of pages16
JournalAnnales Academiae Scientiarum Fennicae Mathematica
Volume44
Issue number1
DOIs
StatePublished - 1 Jan 2019

Keywords

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

ASJC Scopus subject areas

  • Mathematics (all)

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