McShane-Whitney extensions in constructive analysis
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 -continuous functions, where 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.
Cite
@article{arxiv.1804.06757,
title = {McShane-Whitney extensions in constructive analysis},
author = {Iosif Petrakis},
journal= {arXiv preprint arXiv:1804.06757},
year = {2023}
}