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G-strong subdifferentiability and applications to norm attaining subspaces

Javier Falco, Daniel Isert

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.

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.

Paper Structure

This paper contains 10 sections, 32 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminar definitions and properties
  3. James' theorem, Krein-Smulian theorem, and Moreau's maximum formula
  4. Hahn-Banach extension theorem
  5. James' theorem
  6. Moreau's maximum formula
  7. Krein-Smulian theorem
  8. $G$-SSD as a sufficient condition for $G$-reflexivity
  9. Statements and declarations
  10. Competing Interests

Key Result

Proposition 3

Let $X$ be a normed space and $G$ a compact topological group acting on $X$. If $X^{*}_{G} \subsetneq X^{*}$, then the weak group invariant topology on $X$, $w_{G}$, is strictly weaker than the weak topology of $X$, $w$.

Theorems & Definitions (65)

  • Definition 1
  • Definition 2
  • Proposition 3
  • proof
  • proof
  • Proposition 5
  • Theorem 7: Group invariant Banach-Alaoglu-Bourbaki's theorem
  • Definition 8
  • Lemma 9: Helly
  • Theorem 10: Group invariant Goldstine's theorem
  • ...and 55 more