La propiedad de Radon-Nikodym en espacios de Banach duales

• 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
  • Texto completo
• 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.

• Referencias bibliográficas
  • J. Batt and W. Hiermeyer, On compactness in $L\_p(\mu,X)$ in the weak Topology and in the Topology $\sigma(L\_p(\mu,X),L\_q(\mu,X^\*))$, Math....
  • F. Bombal, Sobre los espacios de Orlicz vectoriales, Collectanea Math. 32, 1 (1981), 1–12.
  • J. Bourgain, An averaging result for $\ell^1$-sequences and applications to weakly conditionally compact sets in $L\_X^1$, Israel J. Math....
  • J. K. Brooks and N. Dinculeanu, Weak compactness in Spaces of Bochner integrable functions and applications, Adv. in Math. 24 (1977), 172–188.
  • B. Cascales and G. Vera, Topologies weaker than the weak topology of a Banach space, Aparecerá en J. of Math. Ana. and App.
  • J. Diestel and J. Uhl, Vector measures, Math. Surveys, no 15. Amer. Math. Soc. (1977).
  • G.A. Edgar, Asplund operators and a.e. convergence, J. Multivariate Anal. 10 (1980), 460–466.
  • C. Fierro, Compacidad débil en espacios de funciones integrables-Bochner, y la propiedad de Radon-Nikodým, Rev. Real Acad. Cien. Exact. Fis....
  • K. Floret, Weakly compact sets, LNM 801, Springer-Verlag 1980.
  • N. Ghoussoub and P. Saab, Weak compactness in spaces of Bochner integrable functions and the Radon-Nikodým property, Pacific J. Math. 110,1...
  • J.E. Jayne and C.A. Rogers, Borel selectors for upper semi-continuous set-valued maps, Acta Math. 155 (1985), 41–79.
  • J.E. Jayne, J. Orihuela, A.J. Pallarés and G. Vera, $\alpha$-fragmentability of Multivalued Maps and Selection Theorems, J. Funct. Anal. 117...
  • C. Stegall, The Radon-Nikodým property in conjugate Banach spaces II, Trans. Amer. Math. Soc. 264 (1981), 507–519.
  • H.H. Schaeffer, Topological vector spaces, GTM 3, Springer-Verlag (1980).
  • M. Talagrand, Points extremaux dans le dual de $L^1(\mu,E)$, Proc. Amer. Math. Soc. 99,2 (1984), 265–269.
  • M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. 307, 1984.