McShane-Whitney extensions in constructive analysis

TL;DR

This paper explores constructive versions of the McShane-Whitney extension theorem within Bishop-style mathematics, demonstrating density results and properties of Lipschitz, Hölder, and continuous functions, along with related theorems.

Contribution

It introduces the notion of McShane-Whitney pairs in constructive analysis and extends classical results to this setting, including new extensions for Hölder and continuous functions.

Findings

Lipschitz functions are dense in uniformly continuous functions on totally bounded spaces.

Constructive McShane-Whitney extensions preserve properties of the original functions.

A Lipschitz version of a Hahn-Banach corollary and an approximate extension theorem are established.

Abstract

Within Bishop-style constructive mathematics we study the classical McShane-Whitney theorem on the extendability of real-valued Lipschitz functions defined on a subset of a metric space. Using a formulation similar to the formulation of McShane-Whitney theorem, we show that the Lipschitz real-valued functions on a totally bounded space are uniformly dense in the set of uniformly continuous functions. Through the introduced notion of a McShane-Whitney pair we describe the constructive content of the original McShane-Whitney extension and examine how the properties of a Lipschitz function defined on the subspace of the pair extend to its McShane-Whitney extensions on the space of the pair. Similar McShane-Whitney pairs and extensions are established for H\"{o}lder functions and $\nu$-continuous functions, where $\nu$ is a modulus of continuity. A Lipschitz version of a fundamental corollary of the Hahn-Banach theorem, and the approximate McShane-Whitney theorem are shown.