Equivalent Norms in a Banach Function Space and the Subsequence Property

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 $\Vert f{\chi }_A\Vert \le \Vert f\Vert ,f\in X(\mu ),A\in \Sigma$, 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.

13 pages