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Absolute and Unconditional Convergence in Normed Linear Spaces

Published online by Cambridge University Press:  24 October 2008

D. Rutovitz
D. Rutovitz
Affiliation:
The UniversityManchester
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Extract

In 1933 Orlicz proved various results concerning unconditional convergence in Banach spaces (4), which were noted by Banach ((l), p. 240) who remarked that absolute and unconditional convergence are equivalent in finite dimensional Banach spaces, but that whether or not the two are non-equivalent in all infinite dimensional spaces was still an open question. MacPhail (3) gave a criterion for the equivalence of the two notions of convergence in a general Banach space and used it to prove non-equivalence in the spaces l1 and L1. In 1950 Dvoretzky and Rogers demonstrated the non-equivalence of the two types of convergence in any infinite dimensional normed linear space, using an elegant and instructive geometrical approach (2). The result has also been proved by a different method by Grothendieck (5).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 58 , Issue 4 , October 1962 , pp. 575 - 579
Copyright
Copyright © Cambridge Philosophical Society 1962

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References

REFERENCES

(1)Banach, S., Théorie des opérations linéaires (Warsaw, 1932).Google Scholar
(2)Dvoretzky, A., and Rogers, C. A., Absolute and unconditional convergence in normed linear spaces. Proc. Nat. Acad. Sci. U.S.A. 36 (1950), 192–197.CrossRefGoogle ScholarPubMed
(3)MacPhail, M. S., Absolute and unconditional convergence. Bull. American Math. Soc. 53 (1947), 121–123.Google Scholar
(4)Orliz, W., Über unbedingte Konvergenz in Funktionenräumen (II). Studia Math. 4 (1933), 41–47.Google Scholar
(5)Grothendieck, A., Produits tensoriels topologiques et espaces nucléaires. Mem. American Math. Soc. 16 (1955).Google Scholar