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A note on regular methods of summability and the Banach-Saks property. (English) Zbl 0355.40007

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

MSC:

40C05 Matrix methods for summability
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces

Citations:

Zbl 0276.04003
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