, Volume 31, Issue 1, pp 18-25

Linear problems (with extended range) have linear optimal algorithms

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Abstract

LetF 1 andF 2 be normed linear spaces andS:F 0F 2 a linear operator on a balanced subsetF 0 ofF 1. IfN denotes a finite dimensional linear information operator onF 0, it is known that there need not be alinear algorithmφ:N(F 4) →F 2 which is optimal in the sense that ‖φ(N(f)) −S(f‖ is minimized. We show that the linear problem defined byS andN can be regarded as having a linear optimal algorithm if we allow the range ofφ to be extended in a natural way. The result depends upon imbeddingF 2 isometrically in the space of continuous functions on a compact Hausdorff spaceX. This is done by making use of a consequence of the classical Banach-Alaoglu theorem.