On Weakly Null FDD's in Banach Spaces
Edward Odell, Haskell P. Rosenthal, Thomas Schlumprecht
Abstract
In this paper we show that every sequence (Fn) of finite dimensional subspaces of a real or complex Banach space with increasing dimensions can be ``refined'' to yield an F.D.D. (Gn), still having increasing dimensions, so that either every bounded sequence (xn), with xn in Gn for n in N, is weakly null, or every normalized sequence (xn), with xn in Gn for n in N, is equivalent to the unit vector basis of l1. 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. (Fn), with limn to infty dim (Fn)=infty, so that all normalized sequences (xn), with xn in Fn, n in N, have the same spreading model over X. This spreading model must necessarily be 1-unconditional over X.
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