Equivalent norms in a Banach function space and the subsequence property
J. Korean Math. Soc. 2019 Vol. 56, No. 5, 1387-1401
https://doi.org/10.4134/JKMS.j180682
Published online September 1, 2019
Jose M. Calabuig, Maite Fern\'andez-Unzueta, Fernando Galaz-Fontes, Enrique A. S\'{a}nchez-P\'{e}rez
Universitat Polit\`{e}cnica de Val\`encia; Centro de Investigaci\'on en Matem\'aticas, A.C.; Centro de Investigaci\'on en Matem\'aticas, A.C.; Universitat Polit\`{e}cnica de Val\`encia
Abstract : Consider a finite measure space $(\Ome,\Sig,\mu)$ and a Banach space $X(\mu)$ consisting of (equivalence classes of) real measurable functions defined on $\Ome$ such that $f\chi_A \in X(\mu) $ and $ \|f\chi_A \| \leq \|f\|, \ \pt f \in X(\mu), \ A \in \Sig$. 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.
Keywords : measure space, space of measurable functions, order, Banach function space
MSC numbers : 46E30, 46B42
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