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Inclusions between FK spaces and Kuttner's theorem

Published online by Cambridge University Press:  24 October 2008

I. J. Maddox
I. J. Maddox
Affiliation:
The Queen's University of Belfast
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Extract

The result known as Kuttner's theorem [2] asserts that if 0 < p < 1 and A is a Toeplitz matrix then there is a sequence which is strongly Cesàro summable with index p but which is not A summable. This theorem was extended by Maddox[3] to coregular matrices, and Thorpe [9] gave a further extension by showing that if 0 < p < 1 and X is a locally convex FK space with X ⊃ w0(p) then X ⊃ l∞. Here, w0(p) denotes the space of sequences strongly summable to 0, i.e. x∈w0(p) if and only if

and l∞ denotes the space of bounded sequences. Other proofs of Thorpe's extension and related results appear in Maddox[4, 5].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 3 , May 1987 , pp. 523 - 527
Copyright
Copyright © Cambridge Philosophical Society 1987

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References

REFERENCES

[1]Cooke, R. G.. Infinite Matrices and Sequence Spaces (Macmillan, 1950).Google Scholar
[2]Kuttner, B.. Note on strong summability. J. London Math. Soc. 21 (1946), 118–122.Google Scholar
[3]Maddox, I. J.. On Kuttner's theorem. J. London Math. Soc. 43 (1968), 285–290.CrossRefGoogle Scholar
[4]Maddox, I. J.. FK spaces which include strongly summable sequences. Math. Proc. Cambridge Philos. Soc. 93 (1983), 131–134.CrossRefGoogle Scholar
[5]Maddox, I. J.. Series in locally convex spaces and inclusions between FK spaces. Math. Proc. Cambridge Philos. Soc. 95 (1984), 467–472.CrossRefGoogle Scholar
[6]Maddox, I. J.. Sequence spaces defined by a modulus. Math. Proc. Cambridge Philos. Soc. 100 (1986), 161–166.CrossRefGoogle Scholar
[7]Ruckle, W. H.. FK spaces in which the sequence of coordinate vectors is bounded. Canad. J. Math. 25 (1973), 973–978.CrossRefGoogle Scholar
[8]Snyder, A. K. and Wilansky, A.. Inclusion theorems and semiconservative FK spaces. Rocky Mountain J. Math. 2 (1972), 595–603.CrossRefGoogle Scholar
[9]Thorpe, B.. An extension of Kuttner's theorem. Bull. London Math. Soc. 13 (1981), 301–302.CrossRefGoogle Scholar
[10]Zeller, K.. Allgemeine Eigenschaften von Limitierungsverfahren. Math. Zeit. 53 (1951), 463–487.CrossRefGoogle Scholar