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

TL;DR

The paper addresses Silber's question by showing that a Banach space in which every weakly compact set is super weakly compact must be weakly sequentially complete. It leverages summing sequences (wide-$(s)$ sequences) associated with nontrivial weakly Cauchy sequences to build a weakly compact subset that fails to be super weakly compact, using an explicit witness construction involving the summing functional and associated functionals. The main result demonstrates that the assumption of weak sequential completeness in property (R) is redundant, establishing that X has property (R) if and only if every weakly compact subset of $X$ is super weakly compact. This yields a precise equivalence between weak sequential completeness and the super weakly compactness of weakly compact sets, clarifying the structure of spaces with property (R).

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.

Theorems & Definitions (12)

• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5
• Remark 2.6
• Proposition 2.7
• proof
• Theorem 2.8
• proof