Annals of Pure and Applied Logic 162 (3):182-193 (2010)

Abstract
We discuss computability properties of the set of elements of best approximation of some point xX by elements of GX in computable Banach spaces X. It turns out that for a general closed set G, given by its distance function, we can only obtain negative information about as a closed set. In the case that G is finite-dimensional, one can compute negative information on as a compact set. This implies that one can compute the point in whenever it is uniquely determined. This is also possible for a wider class of subsets G, given that one imposes additionally convexity properties on the space. If the Banach space X is computably uniformly convex and G is convex, then one can compute the uniquely determined point in . We also discuss representations of finite-dimensional subspaces of Banach spaces and we show that a basis representation contains the same information as the representation via distance functions enriched by the dimension. Finally, we study computability properties of the dimension and the codimension map and we show that for finite-dimensional spaces X the dimension is computable, given the distance function of the subspace
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DOI 10.1016/j.apal.2010.09.011
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References found in this work BETA

Unique Solutions.Peter Schuster - 2006 - Mathematical Logic Quarterly 52 (6):534-539.
Borel Complexity and Computability of the Hahn–Banach Theorem.Vasco Brattka - 2008 - Archive for Mathematical Logic 46 (7-8):547-564.
Computational Complexity on Computable Metric Spaces.Klaus Weirauch - 2003 - Mathematical Logic Quarterly 49 (1):3-21.
Corrigendum to “Unique Solutions”.Peter Schuster - 2007 - Mathematical Logic Quarterly 53 (2):214-214.

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