In the recent paper [6], surjective isometries, not necessarily linear, T : AC(X, E) −→ AC(Y, F ) between vector-valued absolutely continuous functions on compact subsets X and Y of the real line, has been described. The target spaces E and F are strictly convex normed spaces. 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 (not necessarily linear) between spaces of absolutely continuous vectorvalued functions, (little) Lipschitz functions and also continuously differentiable functions.2010 Mathematics Subject Classification. Primary 47B38, 47B33, Secondary 46J10.
In this paper, first we study surjective isometries (not necessarily linear) between completely regular subspaces A and B of C 0 (X, E) and C 0 (Y, F ) where X and Y are locally compact Hausdorff spaces and E and F are normed spaces, not assumed to be neither strictly convex nor complete. We show that for a class of normed spaces F satisfying a new defined property related to their T -sets, such an isometry is a (generalized) weighted composition operator up to a translation. Then we apply the result to study surjective isometries between A and B whenever A and B are equipped with certain norms rather than the supremum norm. Our results unify and generalize some recent results in this context.
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