Non linear map preserving Euclidean norm

Let \(V\) be a real vector space endowed with an Euclidean norm \(\Vert \cdot \Vert\).

A bijective map \( T : V \to V\) that preserves inner product \(\langle \cdot, \cdot \rangle\) is linear. Also, Mazur-Ulam theorem states that an onto map \( T : V \to V\) which is an isometry (\( \Vert T(x)-T(y) \Vert = \Vert x-y \Vert \) for all \(x,y \in V\)) and fixes the origin (\(T(0) = 0\)) is linear.

What about an application that preserves the norm (\(\Vert T(x) \Vert = \Vert x \Vert\) for all \(x \in V\))? \(T\) might not be linear as we show with following example:\[
T : & V & \longrightarrow & V \\
& x & \longmapsto & x & \text{if } \Vert x \Vert \neq 1\\
& x & \longmapsto & -x & \text{if } \Vert x \Vert = 1\end{array}\]

It is clear that \(T\) preserves the norm. However \(T\) is not linear as soon as \(V\) is not the zero vector space. In that case, consider \(x_0\) such that \(\Vert x_0 \Vert = 1\). We have:\[
T(2 x_0) &= 2 x_0 \text{ as } \Vert 2 x_0 \Vert = 2\\
T(x_0) + T(x_0) = -x_0 + (-x_0) &= – 2 x_0

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