Dual spaces generated by the interior of the set of norm attaining functionals

Maria D. Acosta ¹   ; Julio Becerra Guerrero ²   ; Manuel Ruiz Galán ²

¹ Departamento de Análisis Matemático Facultad de Ciencias Universidad de Granada 18071 Granada, Spain
² Departamento de Matemática Aplicada E.U. Arquitectura Técnica Universidad de Granada 18071 Granada, Spain

Studia Mathematica, Tome 149 (2002) no. 2, pp. 175-183

Résumé

We characterize some isomorphic properties of Banach spaces in terms of the set of norm attaining functionals. The main result states that a Banach space is reflexive as soon as it does not contain $\ell _1$ and the dual unit ball is the $w^\ast $-closure of the convex hull of elements contained in the “uniform” interior of the set of norm attaining functionals. By assuming a very weak isometric condition (lack of roughness) instead of not containing $\ell _1$, we also obtain a similar result. As a consequence of the first result, a convex-transitive Banach space not containing $\ell _1$ and such that the set of norm attaining functionals has nonempty interior is in fact superreflexive.

Détail

DOI : 10.4064/sm149-2-6

Keywords: characterize isomorphic properties banach spaces terms set norm attaining functionals main result states banach space reflexive soon does contain ell dual unit ball ast closure convex hull elements contained uniform interior set norm attaining functionals assuming weak isometric condition lack roughness instead containing ell obtain similar result consequence first result convex transitive banach space containing ell set norm attaining functionals has nonempty interior superreflexive