Uniform Banach-Saks properties

TL;DR

This paper investigates the uniform Banach-Saks property, establishing its connections with other properties and demonstrating the equivalence of the p-Banach-Saks and strong p-Banach-Saks properties, with applications to the symmetric Kottman constant.

Contribution

It introduces the concept of uniform Banach-Saks property, links it to properties $(A_ fty)$ and the p-Banach-Saks property, and proves the equivalence of the p- and strong p-Banach-Saks properties.

Findings

Established links between uniform Banach-Saks and properties $(A_ fty)$

Proved the equivalence of p-Banach-Saks and strong p-Banach-Saks properties

Applied results to the symmetric Kottman constant

Abstract

The principal aim of this paper is to study the Banach-Saks property when the speed of convergence of a Cesaro mean sequence can be chosen independtly from the choice of the initial sequence. We establish links between the uniform Banach-Saks property, properties $(A_\infty)$ of Partington and the $p$-Banach-Saks property. We also prove that the $p$-Banach-Saks property and the strong $p$-Banach-Saks property are equivalent. Many examples are given and we apply the results to the symmetric Kottman constant.