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Lipschitz-type functions on metric spaces. (English) Zbl 1139.46025

Given a metric space (X,d), let Lip(X) and 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 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)(Y,ρ) between two metric spaces (X,d) and (Y,ρ) is Lipschitz in the small if there exist r>0 and K0 such that ρ(f(x),f(y))K·d(x,y) whenever d(x,y)r. The space (X,d) is called small-determined if LS(X)=Lip(X). Further, (X,d) and (Y,ρ) 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 LS(X) and LS(Y) are isomorphic as unital vector lattices, if and only if Lip * (X) and 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 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:
46E05Lattices of continuous, differentiable or analytic functions
54E35Metric spaces, metrizability
54C35Function spaces (general topology)
54C40Algebraic properties of function spaces (general topology)