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.
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Cobzas, S. (2000). Journal of Mathematical Analysis and Applications, 243, 344–356.
de Blasi, F. S., & Myjak, J. (1998). Journal of Approximation Theory, 94, 54–72.
de Blasi, F. S., Myjak, J., & Papini, P. L. (1991). Journal of the London Mathematical Society, 44, 135–142.
Edelstein, M. (1968). Journal of the London Mathematical Society, 43, 375–377.
Konjagin, S. V. (1978). Soviet Mathematics. Doklady, 19, 309–312.
Kuratowski, C., & Ulam, S. (1932). Fundamenta Mathematicae, 19, 248–251.
Lau, K. S. (1978). Indiana University Mathematics Journal, 27, 791–795.
Li, C. (2000). Journal of Approximation Theory, 107, 96–108.
Reich, S., & Zaslavski, A. J. (2001). Topological Methods in Nonlinear Analysis, 18, 395–408.
Reich, S., & Zaslavski, A. J. (2002). Nonlinear Functional Analysis and Applications, 7, 115–128.
Reich, S., & Zaslavski, A. J. (2003). Journal of Nonlinear and Convex Analysis, 4, 165–173.
Reich, S., & Zaslavski, A. J. (2004). Nonlinear Analysis Forum, 9, 135–152.
Stechkin, S. B. (1963). Revue Roumaine de Mathématiques Pures Et Appliquées, 8, 5–13.
Zaslavski, A. J. (2001). Calculus of Variations and Partial Differential Equations, 13, 265–293.
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Reich, S., Zaslavski, A.J. (2014). Best Approximation. In: Genericity in Nonlinear Analysis. Developments in Mathematics, vol 34. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9533-8_7
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DOI: https://doi.org/10.1007/978-1-4614-9533-8_7
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