This is a problem our teacher gave us and I have a feeling he forgot to mention some additional data.
Let $H$ be a Hilbert space and $T : H \to H$ a everywhere-defined linear operator such that $T$ is self-adjoint, i.e. $\forall x,y \in H : \langle Tx,y\rangle=\langle x,Ty\rangle$.
Show that $T$ is bounded!
Using Cauchy-Schwarz inequality it is clear that if $T$ is idempotent then it's bounded, but with just the information initially provided I don't feel like it's possible to prove the boundedness.
Can somebody shed some light on this? Perhaps with a counter-example?