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arXiv · 1804.06757

McShane-Whitney extensions in constructive analysis

Iosif Petrakis
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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 Hölder functions and $ν$-continuous functions, where $ν$ 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.

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BibTeXRIS

Iosif Petrakis. 2020-02-14. McShane-Whitney extensions in constructive analysis. https://doi.org/10.23638/lmcs-16(1%3A18)2020

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