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Weak compactness in nice Musielak-Orlicz spaces

Mauro Sanchiz

Abstract

We prove two weak compactness criteria in Musielak-Orlicz spaces for $N$-functions satisfying the $Δ_2$-condition. They extend criteria from Andô for Orlicz spaces to this setting of non-symmetrical Banach function spaces. As consequences, we prove criteria for a sequence in a Musielak-Orlicz space to be weakly convergent, and show that Musielak-Orlicz spaces with the subsequence splitting property are weakly Banach-Saks. The study includes the case of Musielak-Orlicz sequence spaces.

Weak compactness in nice Musielak-Orlicz spaces

Abstract

We prove two weak compactness criteria in Musielak-Orlicz spaces for -functions satisfying the -condition. They extend criteria from Andô for Orlicz spaces to this setting of non-symmetrical Banach function spaces. As consequences, we prove criteria for a sequence in a Musielak-Orlicz space to be weakly convergent, and show that Musielak-Orlicz spaces with the subsequence splitting property are weakly Banach-Saks. The study includes the case of Musielak-Orlicz sequence spaces.
Paper Structure (5 sections, 12 theorems, 51 equations)

This paper contains 5 sections, 12 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weak compactness criteria
  4. Weakly convergent sequences
  5. Final comments

Key Result

Theorem 3.1

A subset $S$ in $X$ is relatively $\sigma(X, X')$-compact if and only if the mapping on $X'$ defined by is an absolutely continuous normal semi-norm.

Theorems & Definitions (26)

  • Theorem 3.1: Luxemburg Theorem 5, page 32
  • Theorem 3.2
  • Remark 3.3
  • Definition 3.4
  • Remark 3.5
  • Lemma 3.6
  • proof
  • Remark 3.7
  • Theorem 3.8
  • proof
  • ...and 16 more