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.

This theorem even holds more generally for operators in Banach spaces. See for example Proposition 2.3.11 in Analysis Now by Gert Pedersen.
