ON WEAKLY NULL FDDS IN BANACH-SPACES

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) i s an element of G(n) for n is an element of N, is weakly null, or ever y normalized sequence (x(n)), with x(n) is an element of G(n) for n is an element of N, is equivalent to the unit vector basis of l(1). Cruc ial to the proof are two stabilization results concerning Lipschitz fu nctions on finite dimensional normed spaces. These results also lead t o other applications. We show, for example, that every infinite dimens ional Banach space X contains an F.D.D. (F-n), with lim(n-->infinity) dim(F-n) = infinity, so that all normalized sequences (x(n)), with x(n ) is an element of F-n, n is an element of N, have the same spreading model over X. This spreading model must necessarily be 1-unconditional over X.