Abstract
We present a constructive version of the classical McShane-Whitney theorem on the extendability of real-valued Lipschitz functions defined on a subset of a metric space. Through the introduced notion of a McShane-Whitney pair we study some abstract properties of this extension theorem showing how the behavior of a Lipschitz function defined on the subspace of the pair affect its McShane-Whitney extensions on the space of the pair. As a consequence, a Lipschitz version of the theory around the Hahn-Banach theorem is formed. We work within Bishop’s informal system of constructive mathematics \(\mathrm {BISH}\).
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Notes
- 1.
If \(B \subseteq {\mathbb R}\) is bounded and \(\inf B\) exists, then \(\sup (-B)\) exists and \(\sup (-B) = - \inf B\); if \(m = \inf B\), then by definition m is a lower bound of B and \(\forall _{\epsilon > 0}\exists _{b \in B}(b < m + \epsilon )\), therefore \(-m\) is an upper bound of \(-B\) and \(\forall _{\epsilon> 0}\exists _{-b \in -B}(-b > -m - \epsilon )\). The following constructively provable properties are used in this paper: if \(A, B \subseteq {\mathbb R}\) are inhabited and bounded such that \(\sup A, \inf A, \sup B, \inf B\) exist, then \(\sup (A + B)\) exists and \(\sup (A + B) = \sup A + \sup B\), \(\inf (A + B)\) exists and \(\inf (A + B) = \inf A + \inf B\), if \(\lambda > 0\), then \(\sup (\lambda A), \inf (\lambda A)\) exist and \(\sup (\lambda A) = \lambda \sup A\), \(\inf (\lambda A) = \lambda \inf A\), if \(\lambda < 0\), then \(\sup (\lambda A), \inf (\lambda A)\) exist and \(\sup (\lambda A) = \lambda \inf A\), and \(\inf (\lambda A) = \lambda \sup A\).
- 2.
From this we can explain why it is not constructively acceptable that any pair (X, A) is McShane-Whitney. If \(x_{0} \in {\mathbb R}\) and \(({\mathbb R}, {\mathbb R}x_{0})\) is a McShane-Whitney pair, then by Proposition 8(i) we have that \({\mathbb R}x_{0}\) is located, which implies LPO.
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Petrakis, I. (2017). McShane-Whitney Pairs. In: Kari, J., Manea, F., Petre, I. (eds) Unveiling Dynamics and Complexity. CiE 2017. Lecture Notes in Computer Science(), vol 10307. Springer, Cham. https://doi.org/10.1007/978-3-319-58741-7_33
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