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Integration in Locally Compact Spaces II

Published online by Cambridge University Press:  22 January 2016

Edwin Hewitt  and
H. S. Zuckerman
Edwin Hewitt
Affiliation:
University of Washington, Seattle, Washington, U.S.A.
H. S. Zuckerman
Affiliation:
University of Washington, Seattle, Washington, U.S.A.
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The general problem of producing concrete representations for continuous linear functionals on normed linear spaces, ie., of identifying conjugate spaces, has of course attracted the attention of many mathematicians during the last five decades and has been solved in many cases [1, pp. 59-72]. Likewise, the problem of extending a linear functional defined on a linear subspace of a normed linear space may be regarded as solved by the Hahn-Banach theorem [1, p. 28], although problems involving “natural” extensions, like that yielding the Lebesgue integral from the Riemann integral, remain. In the present paper, we shall consider two “natural’ methods of extending a certain linear functional and show that they are in fact identical. As a by-product, we obtain a concrete representation both for the original functional and for its “natural” extension. In subsequent communications, the writers will consider topologies in certain families of linear functionals, canonical resolutions of linear functionals, and other extension problems.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 3 , October 1951 , pp. 7 - 22
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1951

References

[1] Banach, Stefan. Théorie des opérations linéaires, Monografie Matematyczne, Warszawa, 1932.Google Scholar
[2] Bourbaki, Nicolas. Éléments de mathématique, Livre III, ch. III et IV. Hermann et Cie., Paris, 1942.Google Scholar
[3] Bourbaki, Nicolas, ‘Eléments de mathématique, Livre III, ch. IX. Hermann et Cie., Paris, 1948.Google Scholar
[4] Cartan, Henri. Sur les fondements de la théorie du potentiel. Bull. Soc. Math. France, 69, pp. 71–96 (1941).Google Scholar
[5] Halmos, Paul R. Measure theory. D. Van Nostrand and Co., New York, 1950.Google Scholar
[6] Saks, Stanislas. Theory of the integral. Monografie Matematyczne, Warszawa-Lwow, 1937.Google Scholar
[7] Stone, M. H. Applications of the theory of Boolean rings to general topology. Trans. Amer. Math. Soc, 41, pp. 375–481 (1937).Google Scholar
[8] Hewitt, Edwin. Integration in locally compact spaces I, To appear in Univ. of Washington Studies in Mathematics.Google Scholar