Vector spaces/K/Inner product/Linear isometry/Characterization/Fact

Let ${}V$ and ${}W$ be vector spaces over ${}{\mathbb {K} }$, both endowed with an inner product, and let ${}\varphi \colon V\rightarrow W$ be a linear mapping. Then the following statements are equivalent.

1. ${}\varphi$ is an isometry.
2. For all ${}u,v\in V$, we have ${}d(\varphi (u),\varphi (v))=d(u,v)$.
3. For all ${}v\in V$, we have ${}\Vert {\varphi (v)}\Vert =\Vert {v}\Vert$.
4. For all ${}v\in V$ fulfilling ${}\Vert {v}\Vert =1$, we have ${}\Vert {\varphi (v)}\Vert =1$.

Proof, Write another proof