Computability of Continuous Solutions of Higher-Type Equations

Given a continuous functional $f \colon X \to Y$ and y *** Y , we wish to compute x *** X such that f (x ) = y , if such an x exists. We show that if x is unique and X and Y are subspaces of Kleene---Kreisel spaces of continuous functionals with X exhaustible, then x is computable uniformly in f , y and the exhaustion functional $\forall_X \colon 2^X \to 2$. We also establish a version of the above for computational metric spaces X and Y , where is X computationally complete and has an exhaustible set of Kleene---Kreisel representatives. Examples of interest include functionals defined on compact spaces X of analytic functions.