If $\langle Tx, y \rangle=\langle x, Ty \rangle $ then $T$ is continuous

Question

Let $\mathcal{H}$ be a Hilbert space and $T: \mathcal{H} \to \mathcal{H}$ where $\langle Tx, y \rangle=\langle x, Ty \rangle $ for any $x,y \in \mathcal{H}$.

As a hint, I am supposed to use the Closed Graph Theorem:

So let $(x_{n})_{n}\subseteq \mathcal{H}$ and $x_{n}\xrightarrow{n \to \infty} x$ and $Tx_{n}\xrightarrow{n \to \infty} y$. I need to show $Tx=y$.

The best that I could do so far has been for a null sequence, i.e. $x_{n}\xrightarrow{n \to \infty} 0$ and $Tx_{n} \xrightarrow{n \to \infty} z$

note that:

$\langle z, z \rangle=\lim\limits_{n\to \infty}\langle Tx_{n}, z \rangle=\lim\limits_{n\to \infty}\langle x_{n}, Tz \rangle=0$

But this does not suffice as I need to show it for all sequences not necessarily with a limit $0$, correct?

Answer 1

$\newcommand{\ip}[2]{\langle #1,#2\rangle}$

Hint: To show that $Tx=y$ it's enough to show that $\ip{Tx}z=\ip yz$ for every $z$.