Noncompactness and noncompleteness in isometries of Lipschitz spaces

We solve the following three questions concerning surjective linear isometries between spaces of Lipschitz functions Lip ( X , E ) and Lip ( Y , F ) , for strictly convex normed spaces E and F and metric spaces X and Y:(i) terize those base spaces X and Y for which all isometries are weighted composition maps. condition independent of base spaces under which all isometries are weighted composition maps. e the general form of an isometry, both when it is a weighted composition map and when it is not. rticular, we prove that requirements of completeness on X and Y are not necessary when E and F are not complete, which is in sharp contrast with results known in the scalar context.