On basic sequences in Banach spaces

Abstract

Let X be a Banach space with separable. If X has a shrinking basis and Y is a closed subspace of which contains X, there exists a shrinking basis in X with two complementary subsequences and so that is a reflexive space and , where we are denoting by the weak-star closure of in . If is a sequence in X that converges to a point in for the weak-star topology,there is a basic sequence in such that is a quasi-reflexive Banach space of order one. Given a Banach space Z with basis it is also proved that every basic sequence in Z has a subsequence extending to a basis of Z.