Linear problems (with extended range) have linear optimal algorithms (Q1099022)

Let \(F_ 1\) and \(F_ 2\) be normed linear spaces and \(S: F_ 0\to F_ 2\) a linear operator on a balanced subset \(F_ 0\) of \(F_ 1\). If N denotes a finite dimensional linear information operator on \(F_ 0\), it is known that there need not be a linear algorithm \(\phi\) : N(F\({}_ 4)\to F_ 2\) which is optimal in the sense that \(\| \phi (N(f))- S(f)\|\) is minimized. We show that the linear problem defined by S and N can be regarded as having a linear optimal algorithm if we allow the range of \(\phi\) to be extended in a natural way. The result depends upon imbedding \(F_ 2\) isometrically in the space of continuous functions on a compact Hausdorff space X. This is done by making use of a consequence of the classical Banach-Alaoglu theorem.