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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Keywords: super weakly compact sets, ultrapowers, uniformly convex sets, non linear embeddings in Banach spaces
Mots-clés : Ensembles super faiblement compacts, ultrapuissances, ensembles uniformément convexes, plongements non linéaires dans les espaces de Banach
Lancien, Gilles 1; Raja, Matias 2
CC-BY-ND 4.0
@article{AIF_2022__72_3_1305_0,
author = {Lancien, Gilles and Raja, Matias},
title = {Nonlinear aspects of super weakly compact sets},
journal = {Annales de l'Institut Fourier},
pages = {1305--1328},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {72},
number = {3},
year = {2022},
doi = {10.5802/aif.3488},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3488/}
}
TY - JOUR AU - Lancien, Gilles AU - Raja, Matias TI - Nonlinear aspects of super weakly compact sets JO - Annales de l'Institut Fourier PY - 2022 SP - 1305 EP - 1328 VL - 72 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3488/ DO - 10.5802/aif.3488 LA - en ID - AIF_2022__72_3_1305_0 ER -
%0 Journal Article %A Lancien, Gilles %A Raja, Matias %T Nonlinear aspects of super weakly compact sets %J Annales de l'Institut Fourier %D 2022 %P 1305-1328 %V 72 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3488/ %R 10.5802/aif.3488 %G en %F AIF_2022__72_3_1305_0
Lancien, Gilles; Raja, Matias. Nonlinear aspects of super weakly compact sets. Annales de l'Institut Fourier, Volume 72 (2022) no. 3, pp. 1305-1328. doi : 10.5802/aif.3488. https://aif.centre-mersenne.org/articles/10.5802/aif.3488/
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