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

By the Riesz representation theorem for the dual of C[0;1]C[0;1], for every continuous linear operator F:C[0;1]→RF:C[0;1]→R there is a function g:[0;1]→Rg:[0;1]→R of bounded variation such thatF(f)=∫fdg(f∈C[0;1]). The function g   can be normalized such that V(g)=‖F‖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′S′ mapping F and its norm to some appropriate g.