On basic sequences in Banach spaces


Let X be a Banach space with X<sup>**</sup> separable. If X has a shrinking basis and Y is a closed subspace of X<sup>**</sup> which contains X, there exists a shrinking basis (x<sub>n</sub>) in X with two complementary subsequences (x_{m<sub>i</sub>}) and (x_{n<sub>j</sub>}) so that [x_{m<sub>j</sub>}] is a reflexive space and X +[\widetilde{x_{n<sub>j</sub>}}]= Y, where we are denoting by [\widetilde{x_{n<sub>j</sub>}}] the weak-star closure of [x_{n<sub>j</sub>}] in X<sup>**</sup>. If (y<sub>n</sub>) is a sequence in X that converges to a point in X<sup>**</sup>\thicksim X for the weak-star topology,there is a basic sequence ( y_{n<sub>j</sub>}) in (y<sub>n</sub>) such that [y_{n<sub>j</sub>}] 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<sub>n</sub>) in Z has a subsequence extending to a basis of Z.

DOI Code: 10.1285/i15900932v12p245

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