Normed vector space $l^{2}$

Question

We have a normed vector space $l^{2}$ of Hilbert.

How can we show that

1) if A contained in $l^{2}$ is compact, then A is closed and bounded.

2) in $l^{2}$, there exists a closed and bounded subset that is not compact

Answer 1

It is a basic theorem of Functional Analysis that the closed unit ball of any infinite dimensional normed vector space is not compact. Any book on Functional Analysis has a proof. You can also show directly that $\{e_1,e_2,...\}$ is a closed and bounded set which has no convergent subsequence, hence not compact. Here $e_1=(1,0,0,\cdots ,0),e_2=(0,1,0,\cdots ),\cdots$.