Theorem: Suppose $V$ is a complex vector space and $T:V\longrightarrow V.$ Then
$$\langle{Tv, v}\rangle = 0 \iff T \equiv 0$$
Proof: ($\Leftarrow$) Choose $u,w\in V$. Then $\langle{Tu, w}\rangle = \frac{\langle{T(u+w), u+w}\rangle - \langle{T(u-w), u-w}\rangle}{4} + \frac{\langle{T(u+iw), u+iw}\rangle - \langle{T(u-iw), u-iw}\rangle}{4}i$ which can be verified by computing the RHS. Note that each term on the RHS is of the form $\langle{Tv, v}\rangle$ for appropriate $v\in V$.
My Concern.
I would like to know the motivation behind the algebra with inner products on the RHS. Where does this come from, how can I reproduce it myself, and why is it being used here?
Thanks.