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A Note on Convergence Fields

Published online by Cambridge University Press:  20 November 2018

I. D. Berg
I. D. Berg*
Affiliation:
University of Illinois
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The purpose of this note is to show that the (bounded) convergence field of a conservative matrix is closed under a certain diagonalization procedure.

As an application of the above result we establish a conjecture of Hill and Sledd in (1) and obtain a result of Lorentz originally proved in (2).

First we introduce some notation and definitions, most of which are standard. Let l∞ denote the Banach space of bounded sequences with the supremum norm and let c denote the closed subspace of l∞ consisting of convergent sequences.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 635 - 638
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Hill, J. D. and Sledd, W. T., Summability-(Z, p) and sequences of periodic type, Can. J. Math., 16 (1964), 741–754.Google Scholar
2. Lorentz, G. G., A contribution to the theory of divergent sequences, Acta Math., 80 (1948), 167–190.Google Scholar