Old qual question here:
Suppse $(V,\langle,\rangle)$ is a Hermitian inner product space, and $T:V\to V$ is a complex linear transformation. Show that the following are equivalent:
- For all $v\in V$, $\langle Tv,Tv\rangle=\langle v,v\rangle$
- For all $v,w\in V$, $\langle Tv,Tw\rangle=\langle v,w\rangle$
Now obviously $2\implies1$ but I'm not sure how to go the other way.