G. Grelier | Searcharxiv

arXiv2023

Uniform Banach-Saks properties

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.

math.FA↗

arXiv2022

Lipschitz-free spaces, ultraproducts, and finite representability of metric spaces

We study several properties and applications of the ultrapower $M_{\mathcal U}$ of a metric space $M$. We prove that the Lipschitz-free space $\mathcal F(M_{\mathcal U})$ is finitely representable in $\mathcal F(M)$. We also characterize the metric spaces that are finitely Lipschitz representable in a Banach space as those that biLipschitz embed into an ultrapower of the Banach space. Thanks to this link, we obtain that if $M$ is finitely Lipschitz representable in a Banach space $X$, then $\mathcal F(M)$ is finitely representable in $\mathcal F(X)$. We apply these results to the study of cotype in Lipschitz-free spaces and the stability of Lipschitz-free spaces and spaces of Lipschitz functions under ultraproducts.

math.FA↗