Journal of the
Korean Mathematical Society

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008



J. Korean Math. Soc. 2019; 56(5): 1387-1401

Online first article July 17, 2019      Printed September 1, 2019

Copyright © The Korean Mathematical Society.

Equivalent norms in a Banach function space and the subsequence property

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


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

Supported by: All the authors were supported by Ministerio de Ciencia, Innovaci´on y Universidades (Spain), Agencia Estatal de Investigaciones, and FEDER. J. M. Calabuig and M. Fern´andez-Unzueta under project MTM2014-53009-P. F. Galaz-Fontes under project MTM2009-14483-C02-01 and E. A. S´anchez P´erez under project MTM2016-77054-C2-1-P. M. Fern´andez-Unzueta was also supported by CONACYT 284110.

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