On weakly null FDD'S in Banach spaces
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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 G n for n, is weakly null, or every normalized sequence (x n ),with x n G n for n, is equivalent to the unit vector basis of 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 ndim(F n )=, so that all normalized sequences (x n ),with x n F n, n/, have the same spreading model over X. This spreading model must necessarily be 1-unconditional over X. 1993 The Magnes Press.
