Iosif Petrakis - McShane-Whitney extensions in constructive analysis

lmcs:4455 - Logical Methods in Computer Science, February 17, 2020, Volume 16, Issue 1 - https://doi.org/10.23638/LMCS-16(1:18)2020
McShane-Whitney extensions in constructive analysisArticle

Authors: Iosif Petrakis

    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 $\nu$-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.


    Volume: Volume 16, Issue 1
    Published on: February 17, 2020
    Accepted on: September 22, 2019
    Submitted on: April 24, 2018
    Keywords: Mathematics - Logic
    Funding:
      Source : OpenAIRE Graph
    • Computing with Infinite Data; Funder: European Commission; Code: 731143

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