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The weak Banach-Saks property for Hölder spaces

Prezemysław Górka, Mauro Sanchiz

Abstract

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

The weak Banach-Saks property for Hölder spaces

Abstract

We investigate the weak Banach--Saks property in the setting of Hölder spaces over metric spaces. We show that, for every infinite metric space and every , the Hölder space fails to have the weak Banach--Saks property.
Paper Structure (3 sections, 10 theorems, 39 equations)

This paper contains 3 sections, 10 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main result

Key Result

Proposition 2.1

Let $(M,d)$ be a metric space, and let $(\widehat{M},\widehat{d})$ be its completion. Fix $\alpha \in (0,1]$. Then the following $C^\alpha(\widehat{M}) \cong C^\alpha(M)$ isometric isomorphism holds.

Theorems & Definitions (18)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Proposition 2.5: toland_dual_2020, Corollary 8.11, rewritten
  • Lemma 2.6
  • ...and 8 more