Quantification of the Banach–Saks property

Abstract

We investigate possible quantifications of the Banach-Saks property and the weak Banach-Saks property. We prove quantitative versions of relationships of the Banach-Saks property of a set with norm compactness and weak compactness. We further establish a quantitative version of the characterization of the weak Banach-Saks property of a set using uniform weak convergence and ℓ 1 -spreading models. We also study the case of the unit ball and in this case we prove a dichotomy which is an analogue of the James distortion theorem for ℓ 1 -spreading models.2010 Mathematics Subject Classification. 46B20.

Pełczyński's property () of order p and its quantification

“…Section is inspired by a large number of recent results on quantitative versions of various theorems and properties of Banach spaces. For example, quantitative versions of Krein's theorem were studied in , quantitative versions of Eberlein–S̆mulyan and Gantmacher theorems were investigated in , a quantitative version of James' compactness theorem in , quantifications of weak sequential continuity and of the Schur property in and , quantitative Dunford–Pettis property in , quantification of the Banach–Saks property in , etc. H. Krulišová introduced several possibilities of quantifying Pełczyński's property ( V ).…”

A Quantitative Extension of Szlenk’s Theorem

“…the infimum being taken over all finite subsets A ⊂ N and all sequences of signs ( n ), with n = ±1. Another approach to the quantification of the Banach-Saks properties can be found in the recent paper [7]. The measure γ satisfies the axiomatic definition of a measure of weak noncompactness given in [4].…”

Quantitative Bessaga–Pełczyński property and quantitative Rosenthal property

“…The aim of our paper is to investigate quantitative versions of the above three Banach space properties. Our inspiration comes from many recent quantitative results, such as, quantitative weak sequential completeness and quantitative Schur property [18,20], quantitative Dunford–Pettis property , quantitative reciprocal Dunford–Pettis property , quantitative Banach–Saks property and quantitative Pełczyński's property ( V ) , etc. The present paper is organized as follows.…”