Isometries on certain non-complete vector-valued function spaces

Surjective, not necessarily linear isometries T: AC(X, E)→AC(Y, F) between vector-valued absolutely continuous functions on compact subsets X and Y of the real line have recently been described as generalized weighted composition operators. The target spaces E and F are strictly convex normedspaces. In this paper, we assume that X and Y are compact Hausdorff spaces and E and F are normed spaces, which are not assumed to be strictly convex. We describe (with a short proof) surjective isometries T:(A,‖·‖A)→(B,‖·‖B) between certain normed subspaces A and B of C(X, E)and C(Y, F), respectively. We consider three cases for F with some mild conditions. The first case, in particular, provides a short proof for the above result, without assuming that the target spaces are strictly convex. The other cases give some generalizations in this topic. As a consequence, the results can be applied, for isometries (notnecessarily linear) between spaces of absolutely continuous vector-valued func-tions, (little) Lipschitz functions and also continuously differentiable functions