Abstract: 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) Characterize those base spaces X and Y for which all isometries are weighted composition maps. (ii) Give a condition independent of base spaces under which all isometries are weighted composition maps. (iii) Provide the general form of an isometry, both when it is a weighted composition map and when it is not. In particular, 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.

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