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 ℓ₁ and the dual unit ball is the w*-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 ℓ₁, we also obtain a similar result. As a consequence of the first result, a convex-transitive Banach space not containing ℓ₁ and such that the set of norm attaining functionals has nonempty interior is in fact superreflexive.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-2-6,
author = {Maria D. Acosta and Julio Becerra Guerrero and Manuel Ruiz Gal\'an},
title = {Dual spaces generated by the interior of the set of norm attaining functionals},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {175-183},
zbl = {0998.46007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-2-6}
}
Maria D. Acosta; Julio Becerra Guerrero; Manuel Ruiz Galán. Dual spaces generated by the interior of the set of norm attaining functionals. Studia Mathematica, Tome 151 (2002) pp. 175-183. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm149-2-6/