A simple proof of Singer’s representation theorem

Hensgen, Wolfgang

doi:10.1090/S0002-9939-96-03493-4

A simple proof of Singer’s representation theorem
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by Wolfgang Hensgen PDF

Proc. Amer. Math. Soc. 124 (1996), 3211-3212 Request permission

Abstract:

Let $\Omega$ be a compact Hausdorff space and $X$ a Banach space. Singer’s theorem states that under the dual pairing $(f,m)\mapsto \int \langle f,dm\rangle$, the dual space of $C(\Omega ;X)$ is isometric to $rcabv (\Omega ;X’)$. Using the Hahn-Banach theorem and the (scalar) Riesz representation theorem, a proof of Singer’s theorem is given which appears to be simpler than the proofs supplied earlier by Singer (1957, 1959) and Dinculeanu (1959, 1967).

References

N. V. Prodan, Optimality conditions in problems of linear synthesis, Avtomat. i Telemeh. 10 (1969), 26–34 (Russian, with English summary); English transl., Automat. Remote Control 10 (1969), 1564–1571. MR 0452946
Nicolae Dinculeanu, Sur la représentation intégrale des certaines opérations linéaires. III, Proc. Amer. Math. Soc. 10 (1959), 59–68 (French). MR 104149, DOI 10.1090/S0002-9939-1959-0104149-9
N. Dinculeanu, Vector measures, Hochschulbücher für Mathematik, Band 64, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966. MR 0206189
Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
Ivan Zinger, Linear functionals on the space of continuous mappings of a compact Hausdorff space into a Banach space, Rev. Math. Pures Appl. 2 (1957), 301–315 (Russian). MR 96964
Ivan Singer, Sur les applications linéaires intégrales des espaces de fonctions continues. I, Rev. Math. Pures Appl. 4 (1959), 391–401 (French). MR 115080