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 vector, the statement $(3)$ is clear; so suppose that $v\ne 0$. Then, $\frac{v}{\Vert v\Vert }$ has norm $1$, and, because of

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

we have

$\Vert \phi (v)\Vert =\Vert v\Vert .$

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

To fact