Somewhat quasireflexive Banach spaces
type: Item
instance of: scholarly article
title: Somewhat quasireflexive Banach spaces
zbMATH Open document ID: 0562.46007
DOI: 10.1007/BF02384379
author: Steven F. Bellenot
published in: Arkiv för Matematik
publication date: 1984-01-01T00:00:00Z
review text: One open question in the theory of Banach space is: ''Does each non- reflexive Banach space contain \(c_ 0\), \(\ell_ 1\) or a quasi-reflexive subspace?'' If this question has an affirmative answer, it would imply an affirmative answer to the more famous question: ''Does each infinite dimensional Banach space contain \(c_ 0\), \(\ell_ 1\) or an infinite dimensional reflexive subspace?'' In this paper it is shown that if X is non-reflexive with \(X^{**}/X\) separable then X has ''lots'' of quasi-reflexive subspaces. If \(X^{**}\) is separable, then both X and \(X^*\) have subspaces and quotients which are quasi-reflexive. All these quasi-reflexive subspaces and quotients can be chosen to have a basis. Also obtained is the related result: if \(X^*\) is separable and non- reflexive, then X has a non-reflexive quotient with a shrinking basis. The proofs make heavy use of basis theory.
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 46B15
Mathematics Subject Classification ID: 46B20
zbMATH DE Number: 3895653
zbMATH Keywords: shrinking
zbMATH Keywords: block basic sequences
zbMATH Keywords: non-reflexive Banach space
zbMATH Keywords: quasi- reflexive subspace
zbMATH Keywords: non-reflexive quotient
zbMATH Keywords: shrinking basis
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3895653