On basic sequences in Banach spaces

Abstract

Let X be a Banach space with $X^**$ separable. If X has a shrinking basis and Y is a closed subspace of $X^**$ which contains X, there exists a shrinking basis $(x_n)$ in X with two complementary subsequences $(x_{m_i})$ and $(x_{n_j})$ so that $[x_{m_j}]$ is a reflexive space and $X +[\widetilde{x_{n_j}}]= Y$, where we are denoting by $[\widetilde{x_{n_j}}]$ the weak-star closure of $[x_{n_j}]$ in $X^**$. If $(y_n)$ is a sequence in X that converges to a point in $X^**\thicksim X$ for the weak-star topology,there is a basic sequence $( y_{n_j})$ in $(y_n)$ such that $[y_{n_j}]$ is a quasi-reflexive Banach space of order one. Given a Banach space Z with basis it is also proved that every basic sequence $(z_n)$ in Z has a subsequence extending to a basis of Z.