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

Proof

The implications ${}(1)\Rightarrow (2)$, ${}(2)\Rightarrow (3)$ and ${}(3)\Rightarrow (4)$ are restrictions. ${}(4)\Rightarrow (3)$. For the zero vecto r,the statement ${}(3)$ is clear; so suppose that ${}v\neq 0$. Then, ${}{\frac {v}{\Vert {v}\Vert }}$ has norm ${}1$, and, because of

${}\varphi (v)=\varphi {\left(\Vert {v}\Vert {\frac {v}{\Vert {v}\Vert }}\right)}=\Vert {v}\Vert \varphi {\left({\frac {v}{\Vert {v}\Vert }}\right)}\,,$

we have

${}\Vert {\varphi (v)}\Vert =\Vert {v}\Vert \,.$

${}(3)\Rightarrow (1)$ follows from fact.

To fact