For complex numbers $a_{n}, b_{n}$, what would be the next step to simplify this expression:
$$\left \|\sum_{n=1}^\infty a_{n} b_{n}\right \|^{2}=\left \langle \sum_{n=1}^\infty a_{n}b_{n},\sum_{m=1}^\infty a_{m}b_{m}\right \rangle$$
Is it $=\sum_{n=1}^\infty \sum_{m=1}^\infty |a_{n}|^{2}|b_{m}|^{2}$ !?
where $|a_{n}|^{2}=a_{n}\overline{a_{n}}$