A Characterization of Banach Spaces Containing

A subsequence principle is obtained, characterizing Banach spacescontaining co' in the spirit of the author's 1974 characterization of Banachspaces containing II . Defiaition. A sequence(bj )in a Banach space is called strongly summing (s.s.)if (bj ) is a weak-Cauchy basic sequence so that whenever scalars (cj ) satisfysUPn II E~=I cijll < 00, thenECj converges.A simple permanence property: if (bj ) is an (5.5.) basis for a Banach spaceB and (bj) are its biorthogonal functionals in B* , then (E~=I bj)';1 is anon-trivial weak-Cauchy sequence in B* ; hence B* fails to be weakly sequen-tially complete. (A weak-Cauchy sequence is called non-trivial if it is non-weaklyconvergent. ) Theorem. Every non-trivial weak-Cauchy sequence in a (real or complex) Banachspacehas either an (5.5.) subsequence or a convex block basis equivalent to thesumming basis. Remark. The two alternatives of the theorem are easily seen to be mutuallyexclusive. Corollary 1. A Banach space B contains no isomorph of Co if and only if everynon-trivial weak-Cauchy sequence in B has an (5.5.) subsequence. Combining thecoand II -Theorems, we obtainCorollary 2. 1/ B is a non-reflexive Banach space such that X* is weakly se-quentially complete for all linear subspaces X of B , then Co embeds in B .. infact, Bhas property (u). The proof of the theorem involves a careful study of differences of boundedsemi-continuous functions. The· results of this study may be of independentinterest. DEPARTMENT OF MATHEMATICS, THE UNIVERSITY OF TEXAS AT AUSTIN, AUSTIN, TEXAS 78712-1082E-mail address: rosenth14lmath. utexas • edu License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use