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Phase-isometries between normed spaces

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

Phase-isometry ; Wigner's theorem ; isometry ; real normed space ; projective geometry

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