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Computable Riesz representation for the dual of C [0; 1]
Author(s) -
Lu Hong,
Weihrauch Klaus
Publication year - 2007
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.200710008
Subject(s) - computable analysis , mathematics , bounded function , computability , representation theorem , norm (philosophy) , computable function , operator (biology) , riesz representation theorem , bounded variation , representation (politics) , dual (grammatical number) , operator norm , bounded operator , variety (cybernetics) , discrete mathematics , function (biology) , upper and lower bounds , pure mathematics , mathematical analysis , art , biochemistry , chemistry , statistics , literature , repressor , evolutionary biology , politics , biology , political science , transcription factor , law , gene
By the Riesz representation theorem for the dual of C [0; 1], if F : C [0; 1] → ℝ is a continuous linear operator, then there is a function g : [0;1] → 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 ) = ‖ F ‖. 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 ′ mapping F and its norm to some appropriate g . (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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