Computable Riesz Representation for the Dual of C[0;1]

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Abstract

By the Riesz representation theorem for the dual of C[0;1], for every continuous linear operator F:C[0;1]R there is a function g:[0;1]R of bounded variation such thatF(f)=fdg(fC[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.

Keywords

Computable analysis
integration
Riesz representation theorem

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1

The author has been partially supported by the National Natural Science Foundation of China, NSFC 10 420 130 638 and the Deutsche Forschungsgemeinschaft, DFG: CHV 113/240/0-1.