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Kawamura and Cook have developed a framework for studying the computabilityand complexity theoretic problems over "large" topological spaces. Thisframework has been applied to study the complexity of the differential operatorand the complexity of functionals over the space of continuous functions on theunit interval $C[0,1]$. In this paper we apply the ideas of Kawamura and Cookto the product space of the real numbers endowed with the product topology. Weshow that no computable norm can be defined over such topology. We investigatecomputability and complexity of total functions over the product space in twocases: (1) when the computing machine submits a uniformally bounded number ofqueries to the oracle and (2) when the number of queries submitted by themachine is not uniformally bounded. In the first case we show that the functionover the product space can be reduced to a function over a finite-dimensionalspace. However, in general there exists functions whose computing machines mustsubmit a non-uniform number of queries to the oracle indicating that computingover the product topology can not in general be reduced to computing overfinite-dimensional spaces.

Computability and Complexity over the Product Topology of Real Numbers