A Mazur--Ulam theorem in non-Archimedean normed spaces

TL;DR

This paper extends the Mazur--Ulam theorem to non-Archimedean strictly convex normed spaces, showing that surjective isometries are affine in this setting, unlike the classical case.

Contribution

It establishes a Mazur--Ulam type theorem specifically for non-Archimedean strictly convex normed spaces, filling a gap in the geometric theory.

Findings

Surjective isometries are affine in non-Archimedean strictly convex spaces

Classical Mazur--Ulam theorem does not hold in general for non-Archimedean spaces

New theorem characterizes isometries in this non-Archimedean context

Abstract

The classical Mazur--Ulam theorem which states that every surjective isometry between real normed spaces is affine is not valid for non-Archimedean normed spaces. In this paper, we establish a Mazur--Ulam theorem in the non-Archimedean strictly convex normed spaces.