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 (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 ${c_0}$ is thus obtained, in the spirit of the author’s 1974 subsequence principle characterizing Banach spaces containing ${\ell ^1}$. 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 ${X^{\ast }}$ is weakly sequentially complete for all linear subspaces X of B, then ${c_0}$ embeds in B.