3
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ What does the symbol $(x_n)_n$ mean? Why use two subscripts $n$? $\endgroup$ – Jiaqi Li Dec 15 '17 at 21:15
  • $\begingroup$ @JiaqiLi I modified the notation. $\endgroup$ – Davide Giraudo Dec 15 '17 at 21:20
  • $\begingroup$ Ah, I see. That's much clearer, thanks! $\endgroup$ – Jiaqi Li Dec 15 '17 at 21:34

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.