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<p>We study the reflexivity and strong subdifferentiability within the framework of group invariant mappings. We show that a Banach space is <i>G</i>-reflexive if the norm of its dual is <i>G</i>-strong subdifferentiable. To do so, 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 subspaces inside the set of norm-attaining functionals within a Banach space.</p>

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G-strong subdifferentiability and applications to norm attaining subspaces

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月2日

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  • Javier Falcó,
  • Daniel Isert

摘要

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 so, 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 subspaces inside the set of norm-attaining functionals within a Banach space.