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Summability field

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convergence field of a summation method, for a summation method of sequences (cf. Summation methods)

The set of all sequences summable by the given method. The summability field of any regular matrix summation method (cf. also Regular summation methods) cannot contain all bounded sequences [3]. The summability field of a regular matrix method that sums even one divergent sequence cannot consist of bounded sequences only [4]. However, regular matrix summation methods that sum divergent sequences and whose summability field does not contain bounded divergent sequences do exist. The set of bounded sequences summable by a given method is called the bounded summability field. The summability field of a regular matrix summation method is a complete locally convex metric space with continuous coordinate projections.

References

[1] R.G. Cooke, "Infinite matrices and sequence spaces" , Macmillan (1950)
[2] G.F. Kangro, "Theory of summability of sequences and series" J. Soviet Math. , 5 : 1 (1970) pp. 1–45 Itogi Nauk. i Tekhn. Mat. Anal. , 12 (1974) pp. 5–70
[3] H. Steinhaus, "Some remarks on the generalization of the concept of limit" Prace Mat. Fiz. , 22 (1911) pp. 121–134 (In Polish)
[4] S. Mazur, W. Orlicz, "Sur les méthodes linéaires du sommation" C.R. Acad. Sci. Paris , 196 (1933) pp. 32–34
How to Cite This Entry:
Summability field. I.I. Volkov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Summability_field&oldid=18353
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098