an:00639576
Zbl 0824.46010
Rosenthal, Haskell
A characterization of Banach spaces containing \(c_ 0\)
EN
J. Am. Math. Soc. 7, No. 3, 707-748 (1994).
0894-0347 1088-6834
1994
j
46B03 46B25 03E15
weakly sequentially complete dual; convex block basis; \(\ell_ 1\)- theorem; differences of semi-continuous functions; isomorphic theory of general Banach spaces; characterization of Banach spaces containing \(\ell_ 1\); weak-Cauchy basic sequence; strongly summing subsequence; bounded semi-continuous functions
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: \textit{W. T. Gowers}, A new dichotomy for Banach spaces, preprint; \textit{W. T. Gowers}, Ramsey-type results in Banach space theory, preprint.
M.Ostrovskij (Khar'kov)
0297.46013