Quantification of the Banach-Saks property

TL;DR

This paper develops quantitative measures for the Banach-Saks and weak Banach-Saks properties, linking them with compactness and convergence concepts, and establishes a dichotomy related to $ ext{l}_1$-spreading models.

Contribution

It introduces quantitative versions of key characterizations of the Banach-Saks properties and proves a James distortion type dichotomy for the unit ball.

Findings

Quantitative relationships between Banach-Saks property and compactness.

A quantitative characterization of the weak Banach-Saks property using uniform weak convergence.

A James distortion type dichotomy for $ ext{l}_1$-spreading models on the unit ball.

Abstract

We investigate possible quantifications of the Banach-Saks property and the weak Banach-Saks property. We prove quantitative versions of relationships of the Banach-Saks property of a set with norm compactness and weak compactness. We further establish a quantitative version of the characterization of the weak Banach-Saks property of a set using uniform weak convergence and $\ell_1$-spreading models. We also study the case of the unit ball and in this case we prove a dichotomy which is an analogue of the James distortion theorem for $\ell_1$-spreading models.