Topology. In the $\ell^2$ space. Is the following set compact, bounded and/or closed?

Question

I have been given a problem regarding $\ell^2$ space.

So. Since it's not a finite dimensional problem I know that closed and bounded does not imply compact. I do not have a great deal of experience in dealing with $\ell^2$ space so I think a lot of the confusion stems from that fact.

• I know for closed I need to show that the set contains all limits of its convergent sequences.
• For compact, I'm thinking I can use the fact that sequential compactness is the same thing as compact (in metric spaces).

Any help is greatly appreciated.

Answer 1

Hint: Look at the sequence $\{e_i\}_{i=1}^\infty$ when $e_i=(0,...,0,1,0,0,...)$, the $1$ being in coordinate number $i$. This is a sequence of vectors with norm $1$. Now, what is the norm of $e_i-e_j$ when $i\ne j$? Is there any chance this sequence has a convergent subsequence?