show that $\sum_{n=1}^{\infty}\alpha_n x_n$ converges in $H$ if and only if $\sum_{n=1}^{\infty}|\alpha_n|^2<\infty$.

Question

Let $(H,\left \langle \cdot , \cdot \right \rangle)$ be a Hilbert space and consider a succession $\{x_n\}$ of $H$ such that $\left \langle x_n, x_m \right \rangle=1$ if $n=m$ or $\left \langle x_n, x_m \right \rangle=0$ if $n\neq m$. Prove that $$\sum_{n=1}^{\infty}|\left \langle x, x_m \right \rangle|\leq ||x||^2\text{ for all } x\in H$$ Also, given a succession of scalars $\{\alpha_n\}$, show that $\sum_{n=1}^{\infty}\alpha_n x_n$ converges in $H$ if and only if $\sum_{n=1}^{\infty}|\alpha_n|^2<\infty$.

This question has already been posted here Particular series on Hilbert Space but I still have a question. I have already shown almost everything, I just have to prove that if $\sum_{n=1}^{\infty}\alpha_n x_n$ is convergent, then $\sum_{n=1}^{\infty}|\alpha_n|^2$ is also convergent, how can I do this? Any ideas? Thank you very much for your help.

Answer 1

Let $$ y_n = \sum_{k=1}^{n}\alpha_k x_k $$ $\lim_n y_n$ exists if $\{ y_n \}$ is a Cauchy sequence. Note that \begin{align} \|y_{m+n}-y_{n}\|^2&=\|\sum_{k=m}^{m+n}\alpha_k x_k \|^2 \\ &= \sum_{k=m}^{m+n}|a_k|^2 \\ &= | \sum_{k=1}^{m+n}|\alpha_k|^2-\sum_{k=1}^{m}|\alpha_k|^2| \end{align} So $\{ y_n \}$ is a Cauchy sequence in $H$ iff $\{ \sum_{k=1}^{m}|\alpha_k|^2 \}$ is a Cauchy sequence, which is equivalent to the convergence of $\sum_{k=1}^{\infty}|\alpha_k|^2$.