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

2025/01/28 by Silber, Zdeněk
#46B03 #46B20 #46B50 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2501.16855

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.

Related

• Weak* sequential closures in Banach space theory and their applications
• Grothendieck's compactness principle for the absolute weak topology
• Nonlinear aspects of super weakly compact sets
• On weakly sequentially complete Banach spaces
• Extension of the Erberlein-Smulian Theorem to Normed Spaces