Let $(H, \langle\cdot,\cdot\rangle)$ a Hilbert space and consider a sequence $\{x_n\}_{n\in\mathbb{N}}$ of $H$ such that: $$\langle x_n,x_m\rangle\ =\ \delta_{mn}\ =\ \left\{\begin{array}{ll}1, & n = m\\0, & n\neq m\end{array}\right.$$ Show that $$\sum_{n=1}^{\infty}|\langle x,x_n\rangle|^2\ \leq\ \|x\|^2,\ \forall x\in H.$$ Moreover, given a scalar sequence $\{\alpha_n\}_{n\in\mathbb{N}}$, show that the following are equivalent:
- $\displaystyle\sum_{n=1}^{\infty}\alpha_nx_n$ converge in $H$.
- $\displaystyle\sum_{n=1}^{\infty}|\alpha_n|^2 < +\infty$