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manuscripta mathematica

, Volume 2, Issue 3, pp 285–299 | Cite as

Verallgemeinerung eines Satzes von F. Riesz

  • Edgar Berz
Article

Abstract

Let X be a compact Hausdorff space, C(X) the class of all continuous functions f: X→R, considered as an ordered vector space over R with respect to its canonical order; furthermore let F be an order complete ordered vector space over R. A linear operator A: C(X)→F is called bounded, if it transforms each bounded set of C(X) into a bounded set of F.

The purpose of this paper is to represent such a bounded linear operator A: C(X)→F as an integral Af=∫ dμ with respect to some content μ, defined on the algebraa(X) which is generated by the open sets U⊂X.

This representation is unique, if μ is required to be regular (p. 11).

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Literatur

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    FUCHS, L.: Partially ordered algebraic systems. Pergamon Press, 1963.Google Scholar
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    HALMOS, P. R.: Measure Theory. Van Nostrand, New York, 1966.Google Scholar
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    KAKUTANI, S.: Concrete representation of abstract (M)-spaces. Ann. of Math. (2) 42 994–1024 (1941).Google Scholar
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    KANTOROVITCH, L.: Lineare halbgeordnete Räume. Mat. Sbornik ns. 2 44 (1937).Google Scholar
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    RIESZ, F.: Sur les opérations fonctionelles linéaires. C. R. Acad. Sci. Paris 149, 947–977 (1909)Google Scholar
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    RIESZ, F. et NAGY, B.: Lecons d'analyse fonctionelle. Gauthier-Villars, Paris 1965.Google Scholar
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    SCHAEFER, H.: Topological Vector Spaces. Maxmillan Series, New York, 1966.Google Scholar

Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • Edgar Berz
    • 1
  1. 1.Mathematisches Institut der Universität63 Gießen

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