Let $Q$ be the set of all $x \in H$ of the form $x = \sum_1^{\infty} c_n u_n$ for $|c_n| \le \frac 1n$. Show that $Q$ is compact.

Question

This is Rudin's Real and Complex Analysis Problem 4.6. Let $Q$ be the set of all $x \in H$ of the form $x = \sum_1^{\infty} c_n u_n$ for $|c_n| \le \frac 1n$. Show that $Q$ is compact.

Assume I have a sequence $\{x^k\}$ in which we have $x^k, x^l$ such that

$\|x^k - x^l \|^2 = \sum |c_n^k - c_n^l|^2$

Then I was only able to bound the distance between $x^k, x^l$ with $\frac{2 \pi}{\sqrt{6}}$, a term coming from the $1/n$.

But given arbitrary $\epsilon$, I'm having some troubling finding the subsequence that can lower the difference between the coefficients. Can someone give me some hint?

Answer 1

Hint: It is clearly a closed subset of $H$. Therefore, and since $H$ is complete, your set is complete. Now, prove that it is totally bounded too and you're done.