Erdős, Paul; Magidor, M. A note on regular methods of summability and the Banach-Saks property. (English) Zbl 0355.40007 Proc. Am. Math. Soc. 59, 232-234 (1976). 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 0276.04003)] is used. A Banach space is said to possess Banach-Saks property with respect to \(A\), if every bounded sequence 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 sequence has a subsequence such that each of its subsequences is summable with respect to \(A\). Reviewer: V.Ganapathy Iyer Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 ReviewCited in 23 Documents MSC: 40C05 Matrix methods for summability 46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces Citations:Zbl 0276.04003 PDFBibTeX XMLCite \textit{P. Erdős} and \textit{M. Magidor}, Proc. Am. Math. Soc. 59, 232--234 (1976; Zbl 0355.40007) Full Text: DOI