On quantification of weak sequential completeness

TL;DR

This paper investigates quantitative measures of weak sequential completeness in Banach spaces, providing new inequalities, examples, and partial answers to existing questions, especially in the context of $L$-embedded spaces.

Contribution

It introduces new properties related to weak sequential completeness, improves existing inequalities, and constructs examples demonstrating the limits of these properties.

Findings

Established new inequalities for weak sequential completeness

Constructed a separable Banach space with the Schur property that defies certain renormings

Provided partial answers to open questions in the field

Abstract

We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in $L$-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space $X$ with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.