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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 $ <T^2(v), v> = <Tv, Tv> $ I know $T^2=T^*T $ however I still don't see the jump from $<T^*T(v),v>$ to $<Tv,Tv>$

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.

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By definition of the adjoint operator we have $$\langle T^*x, y\rangle=\langle x, Ty\rangle$$ Now plug in $x=T v$ and $y=v$.

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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.

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$\langle T^*Tv,v\rangle=\langle T^*(Tv),v\rangle=\langle Tv,Tv\rangle$.

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