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Surjective function

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Linear Algebra and its Applications

Let X and Y be real normed spaces and f:X→Y a surjective mapping. Then f satisfies {‖f(x)+f(y)‖,‖f(x)−f(y)‖}={‖x+y‖,‖x−y‖}, x,y∈X, if and only if f is phase equivalent to a surjective linear isometry, that is, f=σU, where U:X→Y is a surjective linear isometry and σ:X→{−1,1}. This is a Wigner's type result for real normed spaces.

Phase-isometries between normed spaces