How to show $\langle T(x),y\rangle=\langle x,S(y)\rangle$ for all $x, y$ implies, S is the adjoint operator? Suppose that $H$ is a Hilbert space and $T$ and $S$ are two functions from $H$ to $H$.
If
$$
\langle T(x),y\rangle=\langle x,S(y)\rangle
$$
for all $x, y \in H$, show that $T$ and $S$ are continuous linear operators with $S=T^*$. The last equality means that $S^*$ is adjoint(Hermitian) operator.
First we need two show $T$ and $S$ are linear operator but we do not have access to the expression representing them in terms of $x$. Also, we need to show they are bounded to conclude they are continuous.
Note: this is not $\mathbb{R}^n$, is a Hilbert space.

Hint: Closed Graph Theorem easily gives continuity of $T$ and $S$. By definition of $T^{*}$ we get $\langle x , T^{*}y \rangle =\langle x , Sy \rangle $ for all $x$ and $y$. Put $x=T^{*} y-Sy$ to see that $\|T^{*} y-Sy\|^{2}=0$ which gives $T^{*} y=Sy$ for all $y$.
Details for continuity: let $x_n \to x$ and $Tx_n \to z$. Then $\langle Tx_n , y \rangle =\langle x_n , Sy \rangle$ and letting $n \to \infty$ we get $\langle z , y \rangle =\langle x , Sy \rangle =\langle Tx , y \rangle$. This is true for all $y$ so $Tx=z$. 
