On a difference between quantitative weak sequential completeness and the quantitative Schur property

TL;DR

This paper investigates quantitative versions of the Schur property and weak sequential completeness in Banach spaces, revealing that these properties can differ quantitatively, with specific examples illustrating this distinction.

Contribution

It demonstrates that the Schur property of  holds in the strongest quantitative form and constructs a Banach space with mixed quantitative properties, highlighting differences between these concepts.

Findings

Schur property of  is quantitatively strongest

Constructed a Banach space with mixed quantitative properties

Showed the quantitative difference between weak sequential completeness and the Schur property

Abstract

We study quantitative versions of the Schur property and weak sequential completeness, proceeding thus with investigations started by G. Godefroy, N. Kalton and D. Li and continued by H. Pfitzner and the authors. We show that the Schur property of $\ell_1$ holds quantitatively in the strongest possible way and construct an example of a Banach space which is quantitatively weakly sequentially complete, has the Schur property but fails the quantitative form of the Schur property.