We consider the existence of closest points in convex subsets of Hilbert sapces. In particular this enables us to define the orthogonal projection onto a closed linear subspace U of a Hilbert space H, and thereby decompose any element x of H as x=u+v, where u is an element of U and v is contained in its orthogonal complement. We also discuss ‘best approximations’ of elements of H in spaces spanned by collections of elements of H.
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