A characterization of Banach spaces containing $c_ 0$

Abstract:

A subsequence principle is obtained, characterizing Banach spaces containing ${c_0}$, in the spirit of the author’s 1974 characterization of Banach spaces containing ${\ell ^1}$. Definition. A sequence $({b_j})$ in a Banach space is called strongly summing (s.s.) if $({b_j})$ is a weak-Cauchy basic sequence so that whenever scalars $({c_j})$ satisfy ${\text {su}}{{\text {p}}_n}\parallel \Sigma _{j = 1}^n{c_j}{b_j}\parallel < \infty$, then $\Sigma {c_j}$ converges. A simple permanence property: if $({b_j})$ is an (s.s.) basis for a Banach space $B$ and $(b_j^ * )$ are its biorthogonal functionals in ${B^ * }$, then $(\Sigma _{j = 1}^nb_j^ * )_{n = 1}^\infty$ is a non-trivial weak-Cauchy sequence in ${B^ * }$; hence ${B^ * }$ fails to be weakly sequentially complete. (A weak-Cauchy sequence is called non-trivial if it is non-weakly convergent.) Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banach space has either an (s.s.) subsequence or a convex block basis equivalent to the summing basis. Remark. The two alternatives of the theorem are easily seen to be mutually exclusive. 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 an (s.s.) subsequence. Combining the ${c_0}$- and ${\ell ^1}$-Theorems, we obtain 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 $B$; in fact, $B$ has property $(u)$. The proof of the theorem involves a careful study of differences of bounded semi-continuous functions. The results of this study may be of independent interest.