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The Hahn-Banach theorem for non-Archimedean-valued fields

Published online by Cambridge University Press:  24 October 2008

A. W. Ingleton
A. W. Ingleton
Affiliation:
King's CollegeNewcastle
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Extract

1. The Hahn-Banach theorem on the extension of linear functionals holds in real and complex Banach spaces, but it is well known that it is not in general true in a normed linear space over a field with a non-Archimedean valuation. Sufficient conditions for its truth in such a space have been given, however, by Monna and by Cohen‡. In the present paper, we show that a necessary condition for the property is that the space be totally non-Archimedean in the sense of Monna, and establish a necessary and sufficient condition on the field for the theorem to hold in every totally non-Archimedean space over the field. This result is obtained as a special case of a more general theorem concerning linear operators, which is analogous to a theorem of Nachbin ((6), Theorem 1) concerning operators in real Banach spaces.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 48 , Issue 1 , January 1952 , pp. 41 - 45
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

REFERENCES

(1)Banach, S.Théorie des opérations linéaires (Warsaw, 1932).Google Scholar
(2)Bohnenblust, H. F. and Sobczyk, A.Extensions of functionals on complex linear spaces. Bull. Amer. math. Soc. 44 (1938), 91–3.CrossRefGoogle Scholar
(3)Cohen, I. S.On non-Archimedean normed spaces. Indag. math. 10 (1948), 244–9.Google Scholar
(4)Kaplansky, I.Maximal fields with valuations. Duke math. J. 9 (1942), 303–21.CrossRefGoogle Scholar
(5)Monna, A. F.Sur les espaces Iinéaires normés. III. Indag. math. 8 (1946), 682–9.Google Scholar
(6)Nachbin, L.A theorem of the Hahn-Banach type for linear transformations. Trans. Amer. math. Soc. 68 (1950), 28–46.CrossRefGoogle Scholar
(7)Ostrowski, A.Unterauchungen zur arithmetischen Theorie der Körper. Math. Z. 39 (1935), 269–404.CrossRefGoogle Scholar