Somewhat quasireflexive Banach spaces
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.
scientific article; zbMATH DE number 3895653