Computability of the Metric Projection Onto Finite-dimensional Linear Subspaces

Authors:
Ruth Dillhage;Vasco Brattka
Affiliations:
Computability & Logic Group, Department of Mathematics and Computer Science, University of Hagen, Hagen, Germany;Laboratory of Foundational Aspects of Computer Science, Department of Mathematics & Applied Mathematics, University of Cape Town, South Africa
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2008

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Abstract

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 @?P"U that maps each finite dimensional linear subspace to the corresponding (single-valued) metric projection is computable.