[Math] Proving that an operator is self-adjoint.

Question

Let $(\lambda_{n})_{n \in \mathbb{N^{*}}}$ be a sequence of real numbers converging to $0$ and let $(u_{n})_{n \in \mathbb{N^{*}}}$ be an orthonormal family in a Hilbert space $H$. Define $T:H \rightarrow H$ by $$T(u) = \sum_{k=1}^{\infty} \lambda_{k}\langle u,u_{k}\rangle u_{k}$$

Prove that $T$ is self-adjoint. could anyone help me please in doing so?

Thanks!

Answer 1

Hint:

$$T\text{ is self adjoint }\Leftrightarrow\langle Tx,x\rangle\in\mathbb{R}\text{ for every }x\in H.$$