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Averaging weakly null sequences. (English) Zbl 0668.46011

Functional analysis, Proc. Semin., Austin/Tex. 1986-87, Lect. Notes Math. 1332, 126-144 (1988).
[For the entire collection see Zbl 0641.00016.]
By a well known result of S. Mazur [Stud. Math. 4, 70-84 (1933; Zbl 0008.31603)] if in a Banach space X, x is the weak-limit of a sequence \((x_ n)\), then it is the norm-limit of a sequence of convex combinations of the original sequence.
The first section of this paper contains a constructive proof yielding repeated averages. The averaging index is generally smaller than the earlier similar indices used by other authors [cf. Z. Zalcwasser, Stud. math. 2, 63-67 (1930); D. C. Gillespie and W. A. Hurwitz, Trans. Am. Math. Soc. 32, 527-543 (1930); W. Szlenk, Stud. Math., 30, 53-61 (1968; Zbl 0169.15303); J. Bourgain, e.g. Bull. Soc. Math. Belgique, Ser. B 31, 87-117 (1979; Zbl 0438.46014)] and in some instances yields more information about the number of averages required.
In §.2 it is shown that in \(C(\omega^{\omega^ n})\), \(n\in {\mathbb{N}}\) every weakly null sequence admits a norm-null convex block subsequence of \(n+1\)-averages. By an example [generalizing the fact discovered by J. Schreier, Stud. Math. 2, 58-62 (1930) that in \(C(\omega^{\omega})\) 1-averages do not suffice] it is shown that this result is the best possible; in particular in \(C(\omega^{\omega^{\omega}})\) it is not always possible to take a bounded number of averages.
Reviewer: J.Benkö

MSC:

46B99 Normed linear spaces and Banach spaces; Banach lattices
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces