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Computable Riesz representation for the dual of C [0; 1]

✍ Scribed by Hong Lu; Klaus Weihrauch

Publisher
John Wiley and Sons
Year
2007
Tongue
English
Weight
308 KB
Volume
53
Category
Article
ISSN
0044-3050

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✦ Synopsis

Abstract

By the Riesz representation theorem for the dual of C [0; 1], if F: C [0; 1] β†’ ℝ is a continuous linear operator, then there is a function g: [0;1] β†’ ℝ of bounded variation such that F (f) = ∫ f d__g__ (f ∈ C [0; 1]). The function g can be normalized such that V (g) = β€–F β€–. In this paper we prove a computable version of this theorem. We use the framework of TTE, the representation approach to computable analysis, which allows to define natural computability for a variety of operators. We show that there are a computable operator S mapping g and an upper bound of its variation to F and a computable operator S β€² mapping F and its norm to some appropriate g. (Β© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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