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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Steinhaus type theorems for summability matrices

Author: I. J. Maddox
Journal: Proc. Amer. Math. Soc. 45 (1974), 209-213
MSC: Primary 40C05; Secondary 40G10
MathSciNet review: 0364938
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Abstract: Necessary and sufficient conditions are given for an infinite matrix to sum all bounded strongly summable sequences. It is shown that the Borel matrix does not sum all such sequences. A corollary is that the bounded summability field of the Borel method is strictly contained in that of the $(C,1)$ method. Also, it is proved that no coregular matrix can almost sum all bounded sequences—a generalization of Steinhaus’ theorem.

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Keywords: Bounded strongly summable sequences, infinite matrix, conservative matrix, Cesàro means, Borel matrix, summability, almost summability
Article copyright: © Copyright 1974 American Mathematical Society