# Prove weighted ball in $l^2$ space is compact

Let $$B \subseteq l^2$$, $$B=\left\{x\in l^2:\sum_{n\geq1}n|x_n|^2\leq1\right\}$$, show that $$B$$ is compact.

My thought: $$B$$ is closed in $$l^2$$ which is complete. Then $$B$$ is complete. It suffices to show $$B$$ is totally bounded. I think we need to first get rid of the infinite tail sum, i.e., bound all sequences with balls centered at sequences that only have finitely many terms. And then find a ball cover for the finite sequences. But I don't know how to bound the tail sum and I'm stuck.

## 1 Answer

Theorem. Let $$p\in[1,+\infty]$$ and $$M\subseteq\ell_p$$ be a bounded subset such that $$\lim\limits_{N\to\infty}\sup\{\Vert (0,0,\ldots,0,x_N,x_{N+1},\ldots) \Vert_p:x\in M\}=0 \tag{1}$$ then $$M$$ is totally bounded.

For the proof, see an answer of that question: How to show that this set is compact in $$\ell^2$$.

The theorem provides a characterization of totally bounded subsets of $$\ell_p$$.

Let’s apply it to $$B$$.

For $$x\in B$$, we have

$$\Vert x\Vert_2 = \sum_{n\ge1} \vert x_n \vert^2 \le \sum_{n\ge1} n\vert x_n \vert^2 \le 1$$ proving that $$B$$ in included in the closed ball centered on the origin with radius equal to one. Hence $$B$$ is bounded.

We’ll be done if we prove that $$B$$ satisfies condition $$(1)$$ of theorem above.

For $$x \in B$$

$$N\sum_{k\ge N} \vert x_k\vert^2 \le \sum_{k\ge N} k\vert x_k\vert^2 \le \sum_{k\ge 1} k\vert x_k\vert^2 \le 1$$ Therefore $$\sup\{\Vert (0,0,\ldots,0,x_N,x_{N+1},\ldots) \Vert_2:x\in B\} \le 1/N$$ and condition $$(1)$$ is satisfied.

$$B$$ is totally bounded. And also complete as being a closed subset of a complete space.

Finally $$B$$ is compact.

• Thank you so much! – Fluffy Skye Jan 17 '19 at 20:25
• @FluffySkye You’re welcome! The ideas that you mentioned in your question are in fact applied behind the stated theorem. – mathcounterexamples.net Jan 17 '19 at 20:27
• @mathcounterexamples.net I think you made a typo in your second last inequality. – BigbearZzz Jan 17 '19 at 20:34
• @BigbearZzz Thanks for noticing. I corrected. – mathcounterexamples.net Jan 17 '19 at 20:35
• I know it's been some time. But how exactly dis you show B is closed? Im trying this excercise be showing that any convergent net in B converges to a point in B... But can't see the trick to make inequalities work – NazimJ Mar 6 '19 at 18:56