I think I have a proof that if $A: H\rightarrow H$ is a self adjoint operator on a Hilbert space $H$, then $A$ is bounded:
We can use the closed graph theorem. Let $x_n \rightarrow x$ and $Ax_n \rightarrow y$. For every $z \in H$, $$\langle z,Ax -y\rangle=\langle z, Ax\rangle -\langle z,y \rangle=\langle Az,x\rangle-\langle z,y\rangle$$ $$=\lim_n \langle Az,x_n\rangle -\lim_n \langle z,Ax_n \rangle=0,$$ by self adjointness. Thus, $Ax=y$.
First of all, is this correct? Second, if it is, then why is there a GTM text titled "Unbounded Self Adjoint Operators on Hilbert Space"? I think it might have to do with the fact that I am assuming here that $A$ is defined on all of $H$, not just on some subspace.
Thanks for the help.