A SUBSEQUENCE PRINCIPLE CHARACTERIZING BANACH SPACES CONTAINING c0

The notion of a strongly summing sequence is introduced. Such a sequence is weak-Cauchy, a basis for its closed linear span, and has the crucial property that the dual of this span is not weakly sequentially complete. The main result is: Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either a strongly summing sequence or a convex block basis equivalent to the summing basis. (A weak-Cauchy sequence is called non-trivial if it is non-weakly convergent.) The following characterization of spaces containing cq is thus obtained, in the spirit of the author's 1974 subsequence principle characterizing Banach spaces containing lx . Corollary 1. A Banach space B contains no isomorph of Cq if and only if every non-trivial weak-Cauchy sequence in B has a strongly summing subsequence. Combining the co-and i '-theorems, one obtains Corollary 2. // B is a non-reflexive Banach space such that X* is weakly sequentially complete for all linear subspaces X of B , then cq embeds in B .