Grothendieck's compactness principle for the absolute weak topology

TL;DR

This paper investigates the properties of absolutely weakly compact sets in Banach lattices, establishing conditions under which they are sequentially compact and characterizing their structure with respect to the positive Schur property.

Contribution

It proves that absolutely weakly compact sets are sequentially compact in Banach lattices and characterizes when they are contained in convex hulls of null sequences, extending Grothendieck's compactness principle.

Findings

Absolutely weakly compact sets are sequentially compact in Banach lattices.

The converse holds if the space is separable or its bidual's unit ball is absolutely weak* compact.

Such sets are contained in convex hulls of null sequences iff the lattice has the positive Schur property.

Abstract

We prove the following results: (i) Every absolutely weakly compact set in a Banach lattice is absolutely weakly sequentially compact. (ii) The converse of (i) holds if $E$ is separable or $B_{E^{**}}$ is absolutely weak$^*$ compact. (iii) Every absolutely weakly compact subset of a Banach lattice is contained in the closed convex hull of an absolutely weakly null sequence if and only if the Banach lattice has the positive Schur property. Examples and applications are provided.