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Best Approximation

  • Simeon Reich
  • Alexander J. Zaslavski
Part of the Developments in Mathematics book series (DEVM, volume 34)

Abstract

In Chap. 7 we study best approximation problems in a general Banach space. It is well known that best approximation problems have solutions only under certain assumptions on the space X. In view of the Lau-Konjagin result these assumptions cannot be removed. On the other hand, many generic results in nonlinear functional analysis hold in any Banach space. Therefore the following natural question arises: can generic results for best approximation problems be obtained in general Banach spaces? In this chapter we answer this question in the affirmative. To this end, we consider a new framework. The main feature of this new framework is that a best approximation problem is determined by a pair consisting of a point and a closed (convex) subset of a Banach space. We consider the complete metric space of such pairs equipped with a natural complete metric and show that for most (in the sense of Baire category) pairs the corresponding best approximation problem has a unique solution. We also provide some generalizations and extensions of this result.

Keywords

Banach Space Natural Number Minimization Problem Closed Subset Weak Topology 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Simeon Reich
    • 1
  • Alexander J. Zaslavski
    • 1
  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

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