Remarks on the set of norm-attaining functionals and differentiability
Volume 241 / 2018
Studia Mathematica 241 (2018), 71-86
MSC: Primary 46B20; Secondary 46B03, 46B26.
DOI: 10.4064/sm8768-6-2017
Published online: 6 November 2017
Abstract
We use the smooth variational principle and a standard renorming to give a short direct proof of the classical Bishop–Phelps–Bollobás theorem on the density of norm-attaining functionals for weakly compactly generated Banach spaces. Then we show that a slight adjustment of a known Preiss–Zajíček differentiability argument provides a simple, useful characterization of individual norms on separable Banach spaces admitting residual sets of norm-attaining functionals in terms of Fréchet differentiability of their dual norms.
