Selecting basic sequences in -stable Banach spaces
Volume 159 / 2003
Abstract
In this paper we make use of a new concept of \varphi -stability for Banach spaces, where \varphi is a function. If a Banach space X and the function \varphi satisfy some natural conditions, then X is saturated with subspaces that are \varphi -stable (cf. Lemma 2.1 and Corollary 7.8). In a \varphi -stable Banach space one can easily construct basic sequences which have a property P(\varphi ) defined in terms of \varphi (cf. Theorem 4.5).
This leads us, for appropriate functions \varphi , to new results on the existence of unconditional basic sequences with some special properties as well as new proofs of some known results. In particular, we get a new proof of the Gowers dichotomy theorem which produces the best unconditionality constant (also in the complex case).
