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Convergence in Sequence Spaces

Published online by Cambridge University Press:  20 January 2009

H. F. Green
H. F. Green
Affiliation:
Department of Mathematics, Battersea College of Technology
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In a perfect sequence space α, on which a norm is defined, we can consider three types of convergence, namely projective convergence, strong projective convergence and distance convergence. In the space σ∞, when distance is defined in the usual way, the last two types of convergence coincide and are distinct from projective convergence ((2), p. 316). In the space σ1 all three types of convergence coincide ((2), p. 316). It will be shown in this paper that, if distance convergence and projective convergence coincide, then all three types of convergence coincide. It will not be assumed that the limit under one convergence is also the limit under the other convergence.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 11 , Issue 2 , November 1958 , pp. 83 - 85
Copyright
Copyright © Edinburgh Mathematical Society 1958

References

REFERENCES

(1) Allen, H. S., Groups of Infinite Matrices, Proc. London Math. Soc. (2), 54 (1952), 111–34.CrossRefGoogle Scholar
(2) Cooke, R. G., Infinite Matrices and Sequence Spaces (Macmillan, London, 1950).Google Scholar
(3) Cooke, R. G., Linear Operators (Macmillan, London, 1953).Google Scholar
(4) Köthe, G., Lösbarkeitsbedingungen für Gleichungen mit unendlich vielen Unbekannten, Journ. für Math., 178 (1938), 193–213.Google Scholar