FUNCTIONS ATTAINING THE SUPREMUM AND ISOMORPHIC PROPERTIES OF A BANACH SPACE D. Acosta, Maria ; Becerra Guerrero, Julio ; Ruiz Galan, Manuel;
We prove that a Banach space that is convex-transitive and such that for some element u in the unit sphere, and for every subspace Μ containing u, it happens that the subset of norm attaining functionals on Μ is second Baire category in is, in fact, almost-transitive and superreflexive. We also obtain a characterization of finite-dimensional spaces in terms of functions that attain their supremum: a Banach space is finite-dimensional if, for every equivalent norm, every rank-one operator attains its numerical radius. Finally, we describe the subset of norm attaining functionals on a space isomorphic to , where the norm is the restriction of a Luxembourg norm on . In fact, the subset of norm attaining functionals for this norm coincides with the subset of norm attaining functionals for the usual norm.m.