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Some logical metatheorems with applications in functional analysis. (English) Zbl 1079.03046
Summary: 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.

MSC:
03F07 Structure of proofs
03F10 Functionals in proof theory
47H10 Fixed-point theorems
03F35 Second- and higher-order arithmetic and fragments
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