Quantitative Dunford-Pettis property

TL;DR

This paper explores the quantitative aspects of the Dunford-Pettis property, revealing that it is inherently quantitative and identifying two stronger, dual versions that are possessed by spaces like L^1 and C(K).

Contribution

It introduces two new stronger, dual forms of the quantitative Dunford-Pettis property and demonstrates their presence in classical function spaces.

Findings

Dunford-Pettis property is automatically quantitative.

L^1 and C(K) spaces have both stronger properties.

Several measures of weak non-compactness coincide in L^1 spaces.

Abstract

We investigate possible quantifications of the Dunford-Pettis property. We show, in particular, that the Dunford-Pettis property is automatically quantitative in a sense. Further, there are two incomparable mutually dual stronger versions of a quantitative Dunford-Pettis property. We prove that $L^1$ spaces and $C(K)$ spaces posses both of them. We also show that several natural measures of weak non-compactness are equal in $L^1$ spaces.