Isometries, Mazur-Ulam theorem and Aleksandrov problem for non-Archimedean normed spaces

摘要

We study isometries between normed spaces over a non-Archimedean valued field K. We show the failure of a MazurUlam theorem in the framework of non-Archimedean spaces. Considering Aleksandrov problem, we prove that a surjective Lipschitz map E→E with the strong distance one preserving property, where E is a finite-dimensional normed space, is an isometry if and only if K is locally compact. We prove also that every isometry E→E for finite-dimensional E is surjective if and only if K is spherically complete and card(k) is finite.