On weak sequential completeness of spaces where weakly compact sets are super weakly compact

TL;DR

This paper proves that Banach spaces where weakly compact sets are super weakly compact are necessarily weakly sequentially complete, answering Silber's 2024 question, by constructing specific weakly compact sets from weakly Cauchy sequences.

Contribution

It establishes a link between super weakly compact sets and weak sequential completeness in Banach spaces, resolving a recent open question.

Findings

Banach spaces with super weakly compact sets are weakly sequentially complete

Construction of weakly compact sets from weakly Cauchy sequences using summing subsequences

Answer to Silber's 2024 question about the structure of such Banach spaces

Abstract

We show that every Banach space in which weakly compact sets are super weakly compact in automatically weakly sequentially complete answering a question by Silber (2024). In the proof we show how to build a weakly compact set which is not super weakly compact from an arbitrary nontrivial weakly Cauchy sequence using the notion of a summing subsequence of Rosenthal or Singer.