Equivalent Norms in a Banach Function Space and the Subsequence Property

TL;DR

This paper characterizes Banach spaces of measurable functions with the subsequence property, showing they are ideals with equivalent norms making them Banach function spaces, and provides a characterization of such norms.

Contribution

It proves that Banach spaces with the subsequence property are ideals with equivalent Banach function space norms, extending the understanding of their structure.

Findings

Spaces with the subsequence property are ideals of measurable functions.

Such spaces admit an equivalent norm making them Banach function spaces.

Provides a characterization of norms equivalent to Banach function space norms.

Abstract

Given a finite measure space $(\Omega,\Sigma,\mu)$, we show that any Banach space $X(\mu)$ consisting of (equivalence classes of) real measurable functions defined on $\Omega$ such that $f \chi_A \in X(\mu) $ and $ \|f \chi_A \| \leq \|f\|, \, f \in X(\mu), \ A \in \Sigma$, and having the subsequence property, is in fact an ideal of measurable functions and has an equivalent norm under which it is a Banach function space. As an application we characterize norms that are equivalent to a Banach function space norm.