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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

General logical metatheorems for functional analysis


Authors: Philipp Gerhardy and Ulrich Kohlenbach
Journal: Trans. Amer. Math. Soc. 360 (2008), 2615-2660
MSC (2000): Primary 03F10, 03F35, 47H09, 47H10
Published electronically: October 5, 2007
MathSciNet review: 2373327
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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

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04429-7
PII: S 0002-9947(07)04429-7
Received by editor(s): March 17, 2006
Published electronically: October 5, 2007
Article copyright: © Copyright 2007 American Mathematical Society