Let $V$ be an inner product space over $\mathbb{R}$, and let $T : V \to V$ be a linear map. I'd like to prove the following statement:
If $\|T(v)\| = \|v\|$ for all $v \in V$, then $\langle u,v \rangle = \langle T(u), T(v)\rangle $ for all $u,v \in V$.
I was advised to consider $\|T(u + v)\|$ but that approach bore no fruit.
Any help would be appreciated!