Boundedness of a linear operator on Hilbert space How can I prove that a linear operator $A$ on a Hilbert space $H$ that satisfies
$$ \langle x,Ay\rangle= \langle y,Ax\rangle$$
for all $x,y\in H$ is bounded (i.e., $\|Ax\|\leq c\|x\| $ for some constant $c>0$)?

Let $B_{x}:y\rightarrow\left<y,Ax\right>$, then $B_{x}$ is a linear operator (beware that one shouldn't take $\left<Ax,y\right>$ because then it is conjugate linear, not linear), and $|B_{x}(y)|=\left|\left<y,Ax\right>\right|=\left|\left<x,Ay\right>\right|\leq\|x\|\|Ay\|$.
For fixed $y$, for all $x$ with $\|x\|\leq 1$, we have $|B_{x}(y)|\leq\|Ay\|$, so 
\begin{align*}
\sup_{\|x\|\leq 1}|B_{x}(y)|\leq\|Ay\|<\infty.
\end{align*}
By Uniform Boundedness Principle we have 
\begin{align*}
\sup_{\|x\|\leq 1}\|B_{x}\|<\infty,
\end{align*}
so
\begin{align*}
\|A\|=\sup\{\left|\left<y,Ax\right>\right|: \|x\|\leq 1, \|y\|\leq 1\}=\sup_{\|x\|\leq 1}\|B_{x}\|<\infty.
\end{align*}
