Recursive characterization of computable real-valued functions and relations

Corresponding to the definition of mu-recursive functions we introduce a class of recursive relations in metric spaces such that each relation is generated from a class of basic relations by a finite number of applications of some specified operators. We prove that our class of recursive relations essentially coincides with our class of densely computable relations, defined via Turing machines. In the special case of the real numbers our subclass of recursive functions coincides with the classical class of computable real-valued functions, defined via Turing machines by Grzegorczyk, Lacombe and others.