Generalized metatheorems on the extractability of uniform bounds in functional analysis

In [6], the second author proved general metatheorems for the extraction of effective uniform bounds from ineffective existence proofs in functional analysis, more precisely from proofs in classical analysis A(:= weakly extensional Peano arithmetic WE-PA in all finite types + quantitifer-free choice + the axiom schema of dependent choice DC) extended with (variants of) abstract bounded metric spaces (X, d), and bounded hyperbolic spaces (X, d, W ) ([6,9,4]) as well as abstract normed linear spaces (X, ‖·‖) with a bounded convex subset C ⊆ X . The theories A[X, d], A [X, d, W ], A[X, ‖ · ‖, C] and further variants – based on CAT(0)-spaces, uniformly convex spaces and inner product spaces – result from extending A to the set T of all finite types over the two ground types 0 and X and adding the necessary constants such as dX and ‖ · ‖X , and (purely universal) axioms for metric, resp. normed linear spaces. In particular, the theories contain an axiom expressing the boundedness of (X, d), resp. the boundedness of the convex subset C. Extending Kohlenbach’s monotone variant of Gödel’s[3] and Spector’s[8] functional (’Dialectica’) interpretation for A to these theories one can extract effective bounds from given ineffective existence proofs, where, using a majorization argument, the extracted bounds are shown to be independent of parameters ranging over the bounded metric space, resp. over the bounded convex subset of the normed linear space. The significance of this rests on the fact that this yields