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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.

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
Full Text: DOI
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