Exercise 16, § 1

Let $E$ be a Hilbert space, $P$ a continuous projector in $E$, that is, a continuous endomorphism of $E$ such that $P^2 = P$. Prove that, for $P$ to be an orthogonal projector, it is necessary and sufficient that $\|P\| \leq 1$. (To see that the condition is sufficient, consider a vector x orthogonal to the subspace kernel of $I - P$.) If $P$ is of finite rank, prove that there exists a closed subspace F of E, of finite codimension, containing $P(E)$ and such that the restriction of $P$ to F is an orthogonal projector.