Prove A=A* where $Ax = \sum_{n}a_{n}\langle x, u_n \rangle u_n =0 $ iff $a_{n} \in \mathbb{R}$

Question

How to prove A=A* where $Ax = \sum_{n}a_{n}\langle x, u_n \rangle u_n=0 $ iff $a_{n} \in \mathbb{R}$ and $[u_n]$ is an orthonormal sequence?

Edit: does it have something to do with the equality: $\langle Ax, y \rangle = \langle x, A^*y \rangle = \overline{\langle A^*y, x \rangle} $?

Answer 1

$$\langle x,Ay\rangle= \langle x,\sum_{n}a_n\langle y, u_n\rangle u_n\rangle =\sum_{n} \overline{a_n\langle y,u_n\rangle}\langle x,u_n\rangle =\sum_{n} \overline{\langle y,\overline{a_n}\langle x,u_n\rangle u_n\rangle} =\sum_{n} \overline{\langle y,a_n\langle x,u_n\rangle u_n\rangle} =\overline{\langle y,Ax \rangle}=\langle Ax,y\rangle$$

$A$ represents a diagonal matrix with all elements real numbers.