Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 363-378

LOCALLY CONVEX SPACES AND SCHUR TYPE PROPERTIES

Saak Gabriyelyan

Ben-Gurion University of the Negev, Department of Mathematics
Beer-Sheva, P.O.\ 653, Israel; saak 'at' math.bgu.ac.il

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 l₁ and (8) every bounded non-precompact sequence in E has a subsequence which is discrete and C-embedded in E_w.

2010 Mathematics Subject Classification: Primary 46A03, 46E10.

Key words: Schur property, weak respecting property, Dunford–Pettis property, sequential Dunford–Pettis property.

Reference to this article: S. Gabriyelyan: Locally convex spaces and Schur type properties. Ann. Acad. Sci. Fenn. Math. 44 (2019), 363-378.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4417