G-strong subdifferentiability and applications to norm attaining subspaces

TL;DR

This paper explores the properties of G-reflexivity and strong subdifferentiability in Banach spaces, extending classical functional analysis concepts to group invariant mappings and applying these to norm-attaining functionals.

Contribution

It introduces the concept of G-strong subdifferentiability, extends key theorems to this framework, and provides conditions for the existence of Banach spaces within norm-attaining functionals.

Findings

G-reflexivity characterized by G-strong subdifferentiability of the dual norm

Extension of classical theorems to group invariant mappings

Conditions for Banach spaces inside norm-attaining functionals

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.