Proving a linear map on a Hilbert space is bounded Let $T$ be a linear map on a Hilbert space H and 
\begin{equation*}
\langle Tx,y\rangle=\langle x, Ty\rangle
\end{equation*}
for all $x,y\in H$. I would like to know why $T$ is bounded. Thank you.

Use the closed graph theorem. I assume you know that $T$ is bounded iff it is continuous at $0$, so we need to show that if $x_n\to 0$, then $Tx_n\to0$. The closed graph theorem (check wiki if you are not familiar with it) allows us to assume that $Tx_n\to y$ for some $y\in H$. Thus we have that for any $x\in H$: $\langle y, x\rangle=\langle \lim_{n\to\infty} Tx_n,x\rangle=\lim_{n\to\infty}\langle Tx_n,x\rangle=\lim_{n\to\infty}\langle x_n,Tx\rangle=\langle\lim_{n\to\infty}x_n,Tx\rangle=\langle0,Tx\rangle=0$.
Now the only vector that is vertical to all $ H $ is $ 0 $, so $ y=0 $ and we are done.
