The weak Banach-Saks property for H\"older spaces

TL;DR

This paper demonstrates that H"older spaces over any infinite metric space do not possess the weak Banach--Saks property, highlighting a limitation in their geometric structure.

Contribution

It establishes that for all infinite metric spaces and all th  , the corresponding H"older spaces lack the weak Banach--Saks property, extending understanding of their geometric properties.

Findings

Hf6lder spaces over infinite metric spaces lack the weak Banach--Saks property.

The result holds for all th   and all infinite metric spaces.

This reveals a fundamental geometric limitation of Hf6lder spaces.

Abstract

We investigate the weak Banach--Saks property in the setting of H\"older spaces over metric spaces. We show that, for every infinite metric space $(M,d)$ and every $\alpha \in (0,1]$, the H\"older space $C^{\alpha}(M)$ fails to have the weak Banach--Saks property.