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January 2011 Nonreflexivity of Banach spaces of bounded harmonic functions on Riemann surfaces
Mitsuru Nakai
Proc. Japan Acad. Ser. A Math. Sci. 87(1): 1-4 (January 2011). DOI: 10.3792/pjaa.87.1

Abstract

We give a simple, short, and easy proof to the Masaoka theorem that if Dirichlet finiteness and boundedness for harmonic functions on a Riemann surface coincide with each other, then the dimension of the linear space of Dirichlet finite harmonic functions on the Riemann surface and the dimension of the linear space of bounded harmonic functions on the Riemann surface are finite and identical. The essence of our proof lies in the observation that the former of the above two Banach spaces is reflexive while the latter is not unless it is of finite dimension.

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Mitsuru Nakai. "Nonreflexivity of Banach spaces of bounded harmonic functions on Riemann surfaces." Proc. Japan Acad. Ser. A Math. Sci. 87 (1) 1 - 4, January 2011. https://doi.org/10.3792/pjaa.87.1

Information

Published: January 2011
First available in Project Euclid: 28 December 2010

zbMATH: 1218.30117
MathSciNet: MR2777229
Digital Object Identifier: 10.3792/pjaa.87.1

Subjects:
Primary: 30F20
Secondary: 30F15 , 30F25 , 46A25

Keywords: Banach space , Dirichlet integral , Green function , Harmonic function , Hilbert space , reflexive

Rights: Copyright © 2011 The Japan Academy

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Vol.87 • No. 1 • January 2011
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