A Subsequence Principle Characterizing

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 c0 is thus obtained, in the spirit of the author’s 1974 subsequence principle characterizing Banach spaces containing l 1. Corollary 1. A Banach space B contains no isomorph of c0 if and only if every non-trivial weak-Cauchy sequence in B has a strongly summing subsequence. Combining the c0-and l1-theorems, one obtains Corollary 2. If B is a non-reflexive Banach space such that X∗ is weakly sequentially complete for all linear subspaces X of B, then c0 embeds in B.