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Algebras which represent their linear functionals

Published online by Cambridge University Press:  24 October 2008

F. F. Bonsall  and
A. W. Goldie
F. F. Bonsall
Affiliation:
King's CollegeDurham UniversityNewcastle upon Tyne
A. W. Goldie
Affiliation:
King's CollegeDurham UniversityNewcastle upon Tyne
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Extract

This paper was originally intended to contain a generalization of a theorem of Banach on the extension of linear functionals. This generalized theorem now appears as a by-product of a study of a class of algebras which we believe to be of much greater interest than the theorem itself. Let X be a vector space over the real field and let π(x) be a sub-additive, positive-homogeneous functional on X. Banach ((2), pp. 27–9) proves that any real linear functional f on a subspace X0 of X which satisfies f(x) ≤ π(x) on X0 can be extended to a real linear functional F on X with F(x) ≤ π(x) on X. One of the essential differences between this theorem and the Hahn-Banach theorem is that π can take negative values.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 49 , Issue 1 , January 1953 , pp. 1 - 14
Copyright
Copyright © Cambridge Philosophical Society 1953

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References

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