G-strong subdifferentiability and applications to norm attaining subspaces

Abstract

We study the reflexivity and strong subdifferentiability within the framework of group invariant mappings. We show that a Banach space is G-reflexive if the norm of its dual is G-strong subdifferentiable. To do this, we extend numerous classical concepts in functional analysis such as weak and weak-star topologies, the polar of a set, duality mapping, to the framework of group invariant mappings. We also extend many classical results in functional analysis including Banach-Alaoglu-Bourbaki's theorem, James' theorem, Moreau's maximum formula, and Krein-Smulian's theorem, to this context. To conclude, we provide an application of these new results by providing sufficient conditions to ensure the existence of closed Banach spaces inside the set of norm-attaining functionals of a Banach space.