Proof mining in $L^p$ spaces

TL;DR

This paper develops a logical framework for $L^p$ spaces, enabling the extraction of computational content from proofs involving these spaces through proof mining techniques.

Contribution

It provides an equivalent implicit characterization and axiomatization of $L^p$ spaces within a higher-order logical system suitable for proof mining.

Findings

Formalization of $L^p$ spaces in positive-bounded logic

A metatheorem for extracting computational content from proofs involving $L^p$ spaces

Application to derive the standard modulus of uniform convexity

Abstract

We obtain an equivalent implicit characterization of $L^p$ Banach spaces that is amenable to a logical treatment. Using that, we obtain an axiomatization for such spaces into a higher-order logical system, the kind of which is used in proof mining, a research program that aims to obtain the hidden computational content of mathematical proofs using tools from mathematical logic. As an aside, we obtain a concrete way of formalizing $L^p$ spaces in positive-bounded logic. The axiomatization is followed by a corresponding metatheorem in the style of proof mining. We illustrate its use with the derivation for this class of spaces of the standard modulus of uniform convexity.