$P$ projector. prove that $\langle Px,x\rangle=\|Px\|^2.$ Let $X$ be a Hilbert space and $P \in B(X)$ a projector. Then for any $x\in X$:
$$\langle Px,x\rangle=\|Px\|^2.$$
My proof: 
$$\|Px\|^{2}=\langle Px,Px\rangle=\langle P^{*}Px,x\rangle=\langle P^2x,x\rangle=\langle Px,x\rangle.$$
Is ok ? Thanks :)

Yes, that is all.
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