Phase-isometries between normed spaces

Abstract

Let $X$ and $Y$ be real normed spaces and $f:X\to Y$ a surjective mapping. Then $f$ satisfies $\{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}$, $x,y\in X$, if and only if $f$ is phase equivalent to a surjective linear isometry, that is, $f=\sigma U$, where $U:X\to Y$ is a surjective linear isometry and $\sigma :X\to \{-1,1\}$. This is a Wigner's type result for real normed spaces.

This is a revised version of the paper From Mazur-Ulam to Wigner