A subsequence principle characterizing Banach spaces containing $c_0$

Abstract

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 \RM(real or complex\RM) Banach space has either a strongly summing sequence or a convex block basis equivalent to the summing basis. (A weak-Cauchy sequence is called {\it non-trivial} if it is {\it non-weakly convergent}.) The following characterization of spaces containing $c_0$ is thus obtained, in the spirit of the author's 1974 subsequence principle. Corollary 1. A Banach space $B$ contains no isomorph of $c_0$ if and only if every non-trivial weak-Cauchy sequence in $B$ has a strongly summing subsequence. Combining the $c_0$-and $\ell^1$-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 $c_0$ embeds in $B$.