Equivalent norms in a banach function space and the subsequence property

Abstract

[EN] Consider a finite measure space (Omega, Sigma, mu) and a Banach space X(mu) consisting of (equivalence classes of) real measurable functions defined on Omega such that f chi(A) is an element of X(mu) and parallel to f chi(A)parallel to <= parallel to f parallel to, for all f is an element of X(mu), A is an element of Sigma. We prove that if it satisfies the subsequence property, then it is 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.