Min-Phase-Isometries and Wigner’s Theorem on Real Normed Spaces

Let X and Y be two real normed spaces. Then a surjective map f : X → Y satisfies min { ‖ f ( x ) + f ( y ) ‖ , ‖ f ( x ) - f ( y ) ‖ } = min { ‖ x + y ‖ , ‖ x - y ‖ } ( x , y ∈ X ) if and only if f is a multiplication of a linear isometry and a map with range { - 1 , 1 } . This can also be considered as a generalization of the famous Wigner’s theorem for real normed spaces.