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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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General logical metatheorems for functional analysis
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by Philipp Gerhardy and Ulrich Kohlenbach PDF
Trans. Amer. Math. Soc. 360 (2008), 2615-2660 Request permission

Abstract:

In this paper we prove general logical metatheorems which state that for large classes of theorems and proofs in (nonlinear) functional analysis it is possible to extract from the proofs effective bounds which depend only on very sparse local bounds on certain parameters. This means that the bounds are uniform for all parameters meeting these weak local boundedness conditions. The results vastly generalize related theorems due to the second author where the global boundedness of the underlying metric space (resp. a convex subset of a normed space) was assumed. Our results treat general classes of spaces such as metric, hyperbolic, CAT(0), normed, uniformly convex and inner product spaces and classes of functions such as nonexpansive, Hölder-Lipschitz, uniformly continuous, bounded and weakly quasi-nonexpansive ones. We give several applications in the area of metric fixed point theory. In particular, we show that the uniformities observed in a number of recently found effective bounds (by proof theoretic analysis) can be seen as instances of our general logical results.
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Additional Information
  • Philipp Gerhardy
  • Affiliation: Department of Mathematics, University of Oslo, Blindern, N-0316 Oslo, Norway
  • Ulrich Kohlenbach
  • Affiliation: Department of Mathematics, Technische Universität Darmstadt, Schlossgarten- straße 7, D-64289 Darmstadt, Germany
  • Received by editor(s): March 17, 2006
  • Published electronically: October 5, 2007
  • © Copyright 2007 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 360 (2008), 2615-2660
  • MSC (2000): Primary 03F10, 03F35, 47H09, 47H10
  • DOI: https://doi.org/10.1090/S0002-9947-07-04429-7
  • MathSciNet review: 2373327