A Banach space dichotomy theorem for quotients of subspaces

Valentin Ferenczi¹

¹Institut de Mathématiques de Jussieu Projet Analyse Fonctionnelle Université Pierre et Marie Curie – Paris 6 Boîte 186, 4, Place Jussieu 75252 Paris Cedex 05, France

Studia Mathematica, Tome 180 (2007) no. 2, pp. 111-131

Résumé

A Banach space $X$ with a Schauder basis is defined to have the restricted quotient hereditarily indecomposable property if $X/Y$ is hereditarily indecomposable for any infinite-codimensional subspace $Y$ with a successive finite-dimensional decomposition on the basis of $X$. The following dichotomy theorem is proved: any infinite-dimensional Banach space contains a quotient of a subspace which either has an unconditional basis, or has the restricted quotient hereditarily indecomposable property.

Détail

DOI : 10.4064/sm180-2-2

Keywords: banach space schauder basis defined have restricted quotient hereditarily indecomposable property hereditarily indecomposable infinite codimensional subspace successive finite dimensional decomposition basis following dichotomy theorem proved infinite dimensional banach space contains quotient subspace which either has unconditional basis has restricted quotient hereditarily indecomposable property