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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A simple proof of Singer's
representation theorem


Author: Wolfgang Hensgen
Journal: Proc. Amer. Math. Soc. 124 (1996), 3211-3212
MSC (1991): Primary 46E15, 46E40
MathSciNet review: 1343697
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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).


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Additional Information

Wolfgang Hensgen
Affiliation: NWF I – Mathematik, Universität Regensburg, D– 93040 Regensburg, Germany
Email: wolfgang.hensgen@mathematik.uni-regensburg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03493-4
PII: S 0002-9939(96)03493-4
Keywords: Vector-valued continuous functions, regular vector measures
Received by editor(s): April 21, 1995
Communicated by: Dale E. Alspach
Article copyright: © Copyright 1996 American Mathematical Society