We introduce an extension of the renowned Wigner’s theorem. For real smooth normed spaces X and Y, with X being strictly convex, we demonstrate that any surjective mapping \(f:X\rightarrow Y\) fulfills the condition \( \max \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\max \{\Vert x+y\Vert ,\Vert x-y\Vert \} \quad (x, y\in X) \) if and only if f is phase-equivalent to a linear isometry. This means that there exists a phase function \(\varepsilon :X\rightarrow \{-1,1\}\) such that the composition \(\varepsilon \cdot f\) is a linear isometry mapping from X onto Y.
