Computability of the Metric Projection Onto Finite-dimensional Linear Subspaces

We show that given a computable Banach space X and a finite-dimensional subspace U of X the set of elements of best approximation of x ∈ X (by elements of U) can be computed as a compact set with negative information. If X is uniformly convex, we can even compute the (unique) element of best approximation. Furthermore, given a uniformly convex computable Banach space X the mapping U ↦ PU that maps each finite dimensional linear subspace to the corresponding (single-valued) metric projection is computable.