Proof Mining in Functional Analysis

Abstract

In recent years (though much influenced by writings of G. Kreisel going back to the 50’s as well as subsequent work by H. Luckhardt and others) an applied form of proof theory systematically evolved which is also called ‘Proof Mining’ ([10], see also [1]). 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. We will present general logical metatheorems ([3,6]) 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. Obviously, certain restrictions on the logical form of the theorems to be proved as well as on the axioms to be used in the proofs are necessary. These restrictions in turn depend on the language of the formal systems used as well as the representation of the relevant mathematical objects such as general function spaces. The correctness of the results, moreover, depends in subtle ways on the amount of extensionality properties used in the proof which has a direct analytic counterpart in terms of uniform continuity conditions.