On weakly null FDD'S 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 ), withx _n ∈G _n forn∈ℕ, is weakly null, or every normalized sequence (x _n ),withx _n ∈G _n forn∈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 spaceX contains an F.D.D. (F _n ),with lim _n→∞dim(F _n )=∞, so that all normalized sequences (x _n ),withx _n ∈F _n ,n∈∕, have the same spreading model overX. This spreading model must necessarily be 1-unconditional overX. Research partially supported by NSF DMS-8903197, DMS-9208482, and TARP 235. Research partially supported by NSF.