Effective Uniform Bounds from Proofs in Abstract Functional Analysis

In recent years (though influenced by papers of G. Kreisel going back to the 50’s, e.g. [75–77]) as well as subsequent work by H. Luckhardt ([83, 84]) and others an applied form of proof theory systematically evolved which is also called ‘Proof Mining’ ([73]). It is concerned with transformations of prima facie ineffective proofs into proofs from which certain quantitative computational information as well as new qualitative information can be read off which was not visible beforehand. Applications have been given in the areas of number theory ([83]), combinatorics [2, 36, 100, 101], algebra ([23–26, 21, 22]) and, most systematically, in the area of functional analysis (see the references below). In particular, general logical metatheorems ([55, 35, 65]) have been proved which guarantee a-priorily for large classes of theorems and proofs in analysis the extractability of effective bounds which are independent from parameters in general classes of metric, hyperbolic and normed spaces if certain local boundedness conditions are satisfied. Unless separability assumptions on the spaces involved are used in a given proof, the independence results from parameters only need metric bounds but no compactness ([35, 65]). The theorems treat results involving concrete Polish metric spaces P (such as IR or C[0, 1]) as well as abstract structures (metric, hyperbolic, normed spaces etc.) which are axiomatically added to the formal systems as kind of ‘Urelements’. It is for the latter structures that we can replace the dependency of the bounds from inputs involving elements of these spaces by hereditary bounds (‘majorants’) of such elements which in our applications will be either natural numbers or number theoretic functions. So we can apply the usual notions of computability and complexity for type-2 functionals and do not have to restrict ourselves to instances of these structures which are representable in some effective way or would carry a computability structure. The latter is only required for the concrete Polish metric spaces where we rely on the usual ‘standard (Cauchy) representation’.