aequationes mathematicae

, Volume 31, Issue 1, pp 18–25

Linear problems (with extended range) have linear optimal algorithms

  • Edward W. Packel
Research Papers

DOI: 10.1007/BF02188168

Cite this article as:
Packel, E.W. Aeq. Math. (1986) 31: 18. doi:10.1007/BF02188168

Abstract

LetF1 andF2 be normed linear spaces andS:F0F2 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.

AMS (1980) subject classification

Primary 41A45, 41A65, 65J10Secondary 46B99, 68C05

Copyright information

© Birkhäuser Verlag 1986

Authors and Affiliations

  • Edward W. Packel
    • 1
  1. 1.Department of Computer ScienceColumbia UniversityNew YorkUSA