Equivalent Norms in a Banach Function Space and the Subsequence Property

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 subsequenc...

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Main Authors Calabuig, Jose M, Unzueta, Maite Fernández, Galaz-Fontes, Fernando, Pérez, Enrique A. Sánchez
Format Journal Article
LanguageEnglish
Published 12.10.2018
Subjects
Mathematics - Functional Analysis
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Summary: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.
DOI:10.48550/arxiv.1810.05714