Exercise 16, § 1

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