Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 355.40007
Autor: Erdös, Paul; Magidor, M.
Title: A note on regular methods of summability and the Banach-Saks property. (In English)
Source: Proc. Am. Math. Soc. 59, 232-234 (1976).
Review: The following theorem is proved. Let A be a regular summability matrix. Then every bounded sequence of elements in the space has a subsequence with the property that either every subsequence of this subsequence is summable by A to one and the same limit or no subsequence of this is summable by A. In the proof a result of F.Galvin and K.Prikry on partion into Borel sets [J. Symb. Logic 38, 193-198 (1973; Zbl 276.04003)] is used. A Banach space is said to possess Banach-Saks property with respect to A, if every bounded sepuence has a summable subsequence. It follows from the result above that if a Banach space has the Banach-Saks property with respect to A, then every bounded sepuence has a subsequence such that each of its subsequences is summable with respect to A.
Reviewer: V.Ganapathy Iyer
Classif.: * 40C05 Matrix methods in summability
46B15 Summability and bases in normed spaces
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