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

 

Some logical metatheorems with applications in functional analysis


Author: Ulrich Kohlenbach
Journal: Trans. Amer. Math. Soc. 357 (2005), 89-128
MSC (2000): Primary 03F10, 03F35, 47H09, 47H10
Published electronically: January 29, 2004
MathSciNet review: 2098088
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Abstract: In previous papers we have developed proof-theoretic techniques for extracting effective uniform bounds from large classes of ineffective existence proofs in functional analysis. Here `uniform' means independence from parameters in compact spaces. A recent case study in fixed point theory systematically yielded uniformity even w.r.t. parameters in metrically bounded (but noncompact) subsets which had been known before only in special cases. In the present paper we prove general logical metatheorems which cover these applications to fixed point theory as special cases but are not restricted to this area at all. Our theorems guarantee under general logical conditions such strong uniform versions of non-uniform existence statements. Moreover, they provide algorithms for actually extracting effective uniform bounds and transforming the original proof into one for the stronger uniformity result. Our metatheorems deal with general classes of spaces like metric spaces, hyperbolic spaces, CAT(0)-spaces, normed linear spaces, uniformly convex spaces, as well as inner product spaces.


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Additional Information

Ulrich Kohlenbach
Affiliation: Department of Computer Science, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark
Email: kohlenb@brics.dk

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03515-9
PII: S 0002-9947(04)03515-9
Keywords: Proof mining, functionals of finite type, convex analysis, fixed point theory, nonexpansive mappings, hyperbolic spaces, CAT(0)-spaces
Received by editor(s): May 12, 2003
Published electronically: January 29, 2004
Additional Notes: The author was partially supported by the Danish Natural Science Research Council, Grant no. 21-02-0474, and BRICS, Basic Research in Computer Science, funded by the Danish National Research Foundation
Article copyright: © Copyright 2004 American Mathematical Society