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Lipschitz-type functions on metric spaces. (English) Zbl 1139.46025
Given a metric space \((X,d)\), let \(\text{Lip}(X)\) and \(\text{Lip}^*(X)\) denote, respectively, the set of all Lipschitz real functions and the set of all bounded Lipschitz real functions on \((X,d)\). In this paper, the authors examine the problem as to when \(\text{Lip}^*(X)\) determines \(X\). To achieve their goal, they introduce several types of Lipschitz functions. In particular, Lipschitz functions in the small and small-determined spaces are found to be useful. Recall that a function \(f:(X,d) \to (Y,\rho)\) between two metric spaces \((X,d)\) and \((Y,\rho)\) is Lipschitz in the small if there exist \(r>0\) and \(K\geq 0\) such that \(\rho(f(x),f(y)) \leq K\cdot d(x,y)\) whenever \(d(x,y)\leq r\). The space \((X,d)\) is called small-determined if \(\text{LS}(X)=\text{Lip}(X)\). Further, \((X,d)\) and \((Y,\rho)\) are called LS-homeomorphic if there exists a homeomorphism \(h\) such that \(h\) and \(h^{-1}\) are Lipschitz in the small.
It is shown that two complete metric spaces \(X\) and \(Y\) are LS-homeomorphic if and only if \(\text{LS}(X)\) and \(\text{LS}(Y)\) are isomorphic as unital vector lattices, if and only if \(\text{Lip}^*(X)\) and \(\text{LS}(Y)\) are isomorphic as either algebras or unital vector lattices. Consequently, in the class of complete small-determined metric spaces \(X\), the Lip-structure of \(X\) is determined by \(\text{Lip}^*(X)\) as an algebra or a unital vector lattice. This is a theorem of the Banach-Stone type. Small-determined metric spaces are LS-homeomorphic invariants, and the class of small-determined spaces includes bounded weakly precompact metric spaces, as well as quasi-convex metric spaces. The authors also investigate properties of small-determined spaces. Among many other things, it is shown that small-determined spaces are precisely those metric spaces on which every uniformly continuous real function can be uniformly approximated by Lipschitz functions.

MSC:
46E05 Lattices of continuous, differentiable or analytic functions
54E35 Metric spaces, metrizability
54C35 Function spaces in general topology
54C40 Algebraic properties of function spaces in general topology
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