Nonlinear aspects of super weakly compact sets
Annales de l'Institut Fourier, Online first, 24 p.

The notion of super weak compactness for subsets of Banach spaces is a strengthening of the weak compactness that can be described as a local version of super-reflexivity. A recent result of K. Tu [32] which establishes that the closed convex hull of a super weakly compact set is super weakly compact has removed the main obstacle to further development of the theory. In this paper we provide a variety of results around super weak compactness in order to show the great scope of this notion. We also give non linear characterizations of super weak compactness in terms of the (non) embeddability of special trees and graphs. We conclude with a few relevant examples of super weakly compact sets in non super-reflexive Banach spaces.

La notion de partie super faiblement compacte d’un espace de Banach est un renforcement de la compacité faible qui peut être décrite comme une version locale de la super réflexivité. Un résultat récent de K. Tu qui assure que l’enveloppe convexe fermée d’un ensemble super faiblement compact est super faiblement compacte a supprimé le principal obstacle au développement de cette théorie. Dans cet article, nous fournissons une variété de résultats autour de la super faible compacité pour montrer la grande portée de cette notion. Nous donnons aussi des caractérisations non linéaires de la super faible compacité en termes de (non) plongement de certains arbres et graphes. Nous concluons avec quelques exemples significatifs de parties super faiblement compactes d’espaces de Banach non super réflexifs.

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DOI: 10.5802/aif.3488
Classification: 46B20,  46B80
Keywords: super weakly compact sets, ultrapowers, uniformly convex sets, non linear embeddings in Banach spaces
Lancien, Gilles 1; Raja, Matias 2

1 Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, CNRS UMR-6623, 16 route de Gray, 25030 Besançon Cédex, Besançon (France)
2 Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Espinardo, Murcia, (Spain)
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Lancien, Gilles; Raja, Matias. Nonlinear aspects of super weakly compact sets. Annales de l'Institut Fourier, Online first, 24 p.

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