Hermetian Inner Product Space Polarization

Question

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:

1. For all $v\in V$, $\langle Tv,Tv\rangle=\langle v,v\rangle$
2. 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.

Answer 1

Hint. Remember the polarization identity: $$\langle v,w\rangle=\frac{1}{4}(\langle v+w,v+w\rangle-\langle v-w,v-w\rangle+i\langle v+iw,v+iw\rangle-i\langle v-iw,v-iw\rangle)$$