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A characterization of Banach spaces containing $ c\sb 0$

Author: Haskell Rosenthal
Journal: J. Amer. Math. Soc. 7 (1994), 707-748
MSC: Primary 46B99; Secondary 46B15, 46B25
MathSciNet review: 1242455
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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.

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Keywords: Weakly sequentially complete dual, convex block basis, the $ {\ell ^1}$-Theorem, differences of semi-continuous functions
Article copyright: © Copyright 1994 American Mathematical Society