I can't seem to derive this results that my book "Linear Algebra Done right" is using without explanation. It must be obvious but I don't see it.
Let $T$ be a self adjoint operator. How do they go from $ \langle T^2(v), v\rangle = \langle Tv, Tv\rangle $
I know $T^2=T^*T $ however I still don't see the jump from $\langle T^*T(v),v\rangle $ to $\langle Tv,Tv\rangle $
Also usually when I read questions/answers with operators and the like they mention Hilbert spaces, but I haven't learned about those at all.
Best Answer
By the definition of the adjoint operator, we have $\langle Tv,w\rangle=\langle v,T^*w\rangle$. Hence $\langle T^2v,w\rangle=\langle Tv,T^*w\rangle$. Since $T$ is self-adjoint, $T^*=T$, and you get your result.