Could you help me solve this problem?
Let $e_1, e_2, ...$ be an orthonormal system in a Hilbert space, $\delta_1, \delta_2 ... \in (0, + \infty)$. Prove that the set of all vectors $\sum _{n=1} ^{\infty} \lambda_n e_n, \ \ \lambda_n \in \mathbb{C}, \ \ |\lambda_n| \le \delta_n \ \ \forall n \in \mathbb{N}$ is compact $\iff$ $\sum_{n=1} ^{\infty}\delta_n^2$ is convergent.
Define $$C_{\mathbf{\delta}}:=\left\{\sum_{n}\lambda_ne_n, \left|\lambda_n\right|\leqslant \delta_n,\sum_{n}\lambda_n^2\lt\infty, \left|\lambda_n\right|\leqslant\delta_n\right\}.$$ If $C_{\delta}$ is compact, then the sequence $\left(x_n\right)_{n\geqslant 1} $ given by $x_n:=\sum_{j=1}^n\delta_je_j$ has a convergent subsequence, hence the sequence $\left(\lVert x_n\rVert^2\right )_{n\geqslant 1} $ is bounded.
For the converse, use this.
