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Biseparating maps between Lipschitz function spaces. (English) Zbl 1169.47024
Let $$X,Y$$ be bounded complete metric spaces and let $$E,F$$ be (real or complex) normed spaces. We write $$\text{Lip}(X,E)= \{$$all bounded $$E$$-valued Lipschitz functions}; $$\text{Lip}(X)= \{$$all bounded Lipschitz functionals}; $$L'(E,F)=\{$$all linear bijections from $$E$$ to $$F\}$$. A map $$T:\text{Lip}(X,E)\to \text{Lip}(Y,F)$$ is said to be separating if $$T$$ is linear and $$\|Tf(y)\|\,\|Tg (y)\|=0$$ for all $$y\in Y$$, whenever $$f,g\in \text{Lip}(X,E)$$ satisfy $$\|fx<\|\|g(x) \|=0$$ for all $$x\in X$$. $$T$$ is said to be biseparating if $$T$$ is bijective and both $$T$$ and $$T^{-1}$$ are separating. The authors establish the following results.
Proposition 1. Let $$T:\text{Lip}(X,E)\to \text{Lip}(Y,F)$$ be a biseparating map. Then there exists a bi-Lipschitz homeomorphism $$h:Y \to X$$ and a map $$J:Y\to L'(E,F)$$ such that $$Tf(y)=(Jy) (f(h(y)))$$ for all $$f\in \text{Lip}(X,E)$$ and $$y\in Y$$.
Proposition 2. Let $$T:\text{Lip}(X)\to \text{Lip}(Y)$$ be a bijective separating map. If $$Y$$ is compact, then $$T$$ is biseparating and continuous.

##### MSC:
 47B38 Linear operators on function spaces (general) 46E10 Topological linear spaces of continuous, differentiable or analytic functions 54C35 Function spaces in general topology
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