2018/09/02 by Mojahedi, Mojtaba, Sady, Fereshteh
#46J10 #47B33 #47B38 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1809.00328
In the recent paper \citeHos, surjective isometries, not necessarily linear, T: \rm AC(X,E) \longrightarrow \rm 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) \longrightarrow (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 vector-valued functions, (little) Lipschitz functions and also continuously differentiable functions.