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

Abstract

We study isometries between normed spaces over a non-Archimedean valued field K . We show the failure of a Mazur–Ulam 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.