On Weakly Null FDD's in Banach Spaces
Abstract
In this paper we show that every sequence (F_n) of finite dimensional subspaces of a real or complex Banach space with increasing dimensions can be ``refined'' to yield an F.D.D. (G_n), still having increasing dimensions, so that either every bounded sequence (x_n), with x_n in G_n for n in N, is weakly null, or every normalized sequence (x_n), with x_n in G_n for n in N, is equivalent to the unit vector basis of l_1. Crucial to the proof are two stabilization results concerning Lipschitz functions on finite dimensional normed spaces. These results also lead to other applications. We show, for example, that every infinite dimensional Banach space X contains an F.D.D. (F_n), with lim_{n to infty} dim (F_n)=infty, so that all normalized sequences (x_n), with x_n in F_n, n in N, have the same spreading model over X. This spreading model must necessarily be 1unconditional over X.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 July 1992
 arXiv:
 arXiv:math/9207207
 Bibcode:
 1992math......7207O
 Keywords:

 Mathematics  Functional Analysis;
 46B