一句话：在 Bishop 式构造性数学中建立 McShane-Whitney 扩张，给出 Lipschitz 函数稠密性及 Hölder/ν-连续扩张。
本文在 Bishop 式构造性数学框架下研究经典 McShane-Whitney 定理，即度量空间子集上实值 Lipschitz 函数能否扩张到全空间。作者采用与经典定理相似的表述，证明全有界空间上的 Lipschitz 实值函数在一致连续函数集中一致稠密。通过引入 McShane-Whitney 对，刻画原扩张的构造性内容，并考察子空间上 Lipschitz 函数的性质如何传递给该对空间上的扩张。类似结果被推广到 Hölder 函数和以连续模 ν 定义的 ν-连续函数。最后给出 Hahn-Banach 定理一个基本推论的 Lipschitz 版本以及近似 McShane-Whitney 定理。
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 $\nu$-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.