## A characterization of Banach spaces containing $$c_ 0$$.(English)Zbl 0824.46010

The present paper is an important contribution to the isomorphic theory of general Banach spaces. It contains a characterization of the class of Banach spaces containing $$c_ 0$$ in the spirit of the author’s [Proc. Nat. Acad. Sci. U.S.A. 71, 2411-2413 (1974; Zbl 0297.46013)] characterization of Banach spaces containing $$\ell_ 1$$. The result requires the following new concept:
Definition. A sequence $$(b_ j)$$ in a Banach space is called strongly summing if $$(b_ j)$$ is a weak-Cauchy basic sequence so that whatever scalars $$(c_ j)$$ satisfy $$\sup_ n \left\| \sum^ n_{j= 1} c_ j b_ j\right\|< \infty$$, then $$\sum c_ j$$ converges. A weak-Cauchy sequence is called non-trivial if it is non-weakly convergent.
The author proved: A Banach space $$B$$ contains no isomorph of $$c_ 0$$ if and only if every non-trivial weak-Cauchy sequence in $$B$$ has a strongly summing subsequence.
The paper contains results on bounded semi-continuous functions which are of independent interest.
Reviewer’s remark: Here are references to some of other outstanding recent achievements in the isomorphic theory of general Banach spaces: W. T. Gowers, A new dichotomy for Banach spaces, preprint; W. T. Gowers, Ramsey-type results in Banach space theory, preprint.

### MSC:

 46B03 Isomorphic theory (including renorming) of Banach spaces 46B25 Classical Banach spaces in the general theory 03E15 Descriptive set theory

Zbl 0297.46013
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