## A characterization of Banach spaces containing $c_ 0$

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- by Haskell Rosenthal
- J. Amer. Math. Soc.
**7**(1994), 707-748 - DOI: https://doi.org/10.1090/S0894-0347-1994-1242455-4
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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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*A space not containing*${c_0},\;{\ell _1}$

*or a reflexive subspace*, preprint.

*Differences of bounded semi-continuous functions*(in preparation). —,

*Boundedly complete weak-Cauchy basic sequences in Banach spaces with the PCP*(to appear).

## Bibliographic Information

- © Copyright 1994 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**7**(1994), 707-748 - MSC: Primary 46B99; Secondary 46B15, 46B25
- DOI: https://doi.org/10.1090/S0894-0347-1994-1242455-4
- MathSciNet review: 1242455