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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.

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