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