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

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

DOI: https://doi.org/10.1090/S0002-9939-96-03493-4

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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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