La propiedad de Radon-Nikodym en espacios de Banach duales
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Autores: Antonio José Pallarés Ruiz
, Bernardo Cascales Salinas
- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 45, Fasc. 3, 1994, págs. 263-270
- Idioma: español
- Enlaces
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Resumen
In this note we refine some classical characterizations of the Radon-Nikodým property (briefly RNP) for dual Banach spaces. We prove that the dual space $X^\ast$, of a given Banach space $(X,\parallel\mbox{ }\parallel)$, has the RNP if, and only if, for every probability space ($\Omega,\Sigma,\mu)$ and for every $\mu$-Bochner measurable function $f :\Omega\rightarrow X$ there exists a $\mu$-Bochner measurable function $g : \Omega\rightarrow X^\ast$ such that $\parallel f(\omega)\parallel =< g(\omega), f(\omega) >$ for every $\omega$ in $\Omega$. In the process we point out that spaces of $X$-valued Bochner integrable functions have properties similar to those of spaces of scalar integrable functions if, and only if, the RNP holds inthe dual $X^\ast$ of the range Banach space. We also show what is requiered for a Banach space not containing $\ell^1$ to have a dual with the RNP.
