Let be bounded complete metric spaces and let be (real or complex) normed spaces. We write all bounded -valued Lipschitz functions}; all bounded Lipschitz functionals}; all linear bijections from to . A map is said to be separating if is linear and for all , whenever satisfy for all . is said to be biseparating if is bijective and both and are separating. The authors establish the following results.
Proposition 1. Let be a biseparating map. Then there exists a bi-Lipschitz homeomorphism and a map such that for all and .
Proposition 2. Let be a bijective separating map. If is compact, then is biseparating and continuous.