Dual spaces generated by the interior of the set of norm attaining functionals
Volume 149 / 2002
Studia Mathematica 149 (2002), 175-183
MSC: 46B10, 46B04.
DOI: 10.4064/sm149-2-6
Abstract
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.
