Why is $\langle Ax, Ax \rangle = \langle A^2 x, x\rangle$?

Question

Let $X$ be a Hilbert space and $A\in \mathcal{B}(X)$ be self-adjoint. How can I prove: $$\langle Ax, Ax \rangle = \langle A^2 x, x \rangle$$

I know it is a simple problem, but I don't know how to prove it.

Thanks for your help.

Answer 1

Since $A$ is symmetric or self-adjoint, just use the $\color{red}{\text{definition}}$: $\langle A^2x, x\rangle = \langle A(Ax),x\rangle\color{red}{=}\langle Ax, Ax\rangle$.