A characterization of Banach spaces containing $c_ 0$

Author:
Haskell Rosenthal

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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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*Weakly null normalized sequences in Banach spaces*, Doctoral Thesis, Yale University, 1978.

*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).

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Additional Information

Keywords:
Weakly sequentially complete dual,
convex block basis,
the <IMG WIDTH="23" HEIGHT="23" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="${\ell ^1}$">-Theorem,
differences of semi-continuous functions

Article copyright:
© Copyright 1994
American Mathematical Society