Locally convex spaces and Schur type properties

TL;DR

This paper extends Rosenthal's characterization of Banach spaces with the Schur property to certain locally convex spaces, establishing equivalences involving compactness and sequence properties.

Contribution

It generalizes the Schur property characterization from Banach spaces to quasi-complete locally convex spaces with metrizable separable bounded sets.

Findings

Equivalence of Schur property with matching compact sets in E and E_w

Characterization of sequences in E related to $ ext{ell}_1$ basis

Conditions for sequences to be discrete and C-embedded in E_w

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 $\ell_1$ and (8) every bounded non-precompact sequence in $E$ has a subsequence which is discrete and $C$-embedded in $E_w$.