Locally convex spaces and Schur type properties

2018/01/06 by Gabriyelyan, Saak
#46A03 #FOS: Mathematics #Functional Analysis (math.FA) #General Topology (math.GN)

paper · doi:10.48550/arxiv.1801.01992

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.

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