Phase-isometries between normed spaces

Let X and Y be real normed spaces and f : X -> Y a surjective mapping. Then f satisfies {parallel to f (x) + f (y)parallel to, parallel to f (x) - f(y)parallel to} = {parallel to x + y parallel to, parallel to x - y parallel to}, x,y is an element of X, if and only if f is phase equivalent to a surjective linear isometry, that is, f = sigma U, where U: X -> Y is a surjective linear isometry and sigma: X -> {-1, 1}. This is a Wigner's type result for real normed spaces. (C) 2020 Elsevier Inc. All rights reserved.