Geodesic
Functional Analysis
A class of Banach spaces with no unconditional basic sequence
[Une classe d'espace de Banach sans suite basique inconditionnelle]

Présenté par : Michel Talagrand

Argyros, Spiros A.1 ; Lopez-Abad, Jordi2 ; Todorcevic, Stevo3
1 Department of Mathematics, National Technical University of Athens, Zogratou Campus, 15780 Athens, Greece
2 Équipe de logique mathématique, Université Paris VII, 2, place Jussieu, 75251 Paris cedex, France
3 CNRS–Université Paris VII, 2, place Jussieu, 75251 Paris cedex, France
Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 43-48.

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We give a construction of a reflexive Banach space Xω1 with a transfinite basis of length ω1 and with no unconditional basic sequence. In addition every bounded operator from a subspace of Xω1 into the space Xω1 is a sum of a simple diagonal operator and a strictly singular one.

Nous construisons un espace de Banach réflexif Xω1 ayant une base transfinie de longueur ω1 et n'admettant aucune suite basique inconditionnelle. De plus, tout opérateur borné d'un sous-espace de Xω1 dans cet espace est somme d'un opérateur diagonal très simple et d'un opérateur strictement singulier.

Reçu le :
Accepté le :
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DOI : 10.1016/S1631-073X(03)00272-3

Argyros, Spiros A. 1 ; Lopez-Abad, Jordi 2 ; Todorcevic, Stevo 3

1 Department of Mathematics, National Technical University of Athens, Zogratou Campus, 15780 Athens, Greece
2 Équipe de logique mathématique, Université Paris VII, 2, place Jussieu, 75251 Paris cedex, France
3 CNRS–Université Paris VII, 2, place Jussieu, 75251 Paris cedex, France 
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Argyros, Spiros A.; Lopez-Abad, Jordi; Todorcevic, Stevo. A class of Banach spaces with no unconditional basic sequence. Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 43-48. doi : 10.1016/S1631-073X(03)00272-3. http://geodesic.mathdoc.fr/articles/10.1016/S1631-073X(03)00272-3/

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