On a Question of Bouras concerning weak compactness of almost Dunford-Pettis sets

TL;DR

This paper proves that in Banach lattices, almost Dunford-Pettis sets are weakly compact precisely when the space is a KB-space, answering a question posed by Bouras.

Contribution

It establishes a necessary and sufficient condition linking weak compactness of almost Dunford-Pettis sets to KB-spaces in Banach lattices.

Findings

Almost Dunford-Pettis sets are weakly compact iff the space is a KB-space.

Provides a complete characterization of weak compactness in this context.

Answers an open question by Bouras.

Abstract

We give a positive answer to the question of K. Bouras [`Almost Dunford-Pettis sets in Banach lattices', \textit{Rend. Circ. Mat. Palermo (2)} \textbf{ 62} (2013), 227--236] concerning weak compactness of almost Dunford-Pettis sets in Banach lattices. That is, every almost Dunford-Pettis set in a Banach lattice $E$ is relatively weakly compact if, and only if, $E$ is a $KB$-space.