Linear problems (with extended range) have linear optimal algorithms
- Cite this article as:
- Packel, E.W. Aeq. Math. (1986) 31: 18. doi:10.1007/BF02188168
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LetF1 andF2 be normed linear spaces andS:F0 →F2 a linear operator on a balanced subsetF0 ofF1. IfN denotes a finite dimensional linear information operator onF0, it is known that there need not be alinear algorithmφ:N(F4) →F2 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 imbeddingF2 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.