Petrakis, Iosif
(2020):
MCSHANEWHITNEY EXTENSIONS IN CONSTRUCTIVE ANALYSIS.
In: Logical Methods in Computer Science, Vol. 16, No. 1, 18

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Abstract
Within Bishopstyle constructive mathematics we study the classical McShaneWhitney theorem on the extendability of realvalued Lipschitz functions defined on a subset of a metric space. Using a formulation similar to the formulation of McShaneWhitney theorem, we show that the Lipschitz realvalued functions on a totally bounded space are uniformly dense in the set of uniformly continuous functions. Through the introduced notion of a McShaneWhitney pair we describe the constructive content of the original McShaneWhitney extension and we examine how the properties of a Lipschitz function defined on the subspace of the pair extend to its McShaneWhitney extensions on the space of the pair. Similar McShaneWhitney pairs and extensions are established for Holder functions and nucontinuous functions, where nu is a modulus of continuity. A Lipschitz version of a fundamental corollary of the HahnBanach theorem, and the approximate McShaneWhitney theorem are shown.