My question is: How can I see if (in $\mathcal H=\mathcal l (\mathbb{N} )$
$B_1=\left\{ u | \frac{|u_k|}{k^2}\leq1 \right \}$ ,$B_2=\left\{ u | \frac{|u_k|}{log(1+k)}\leq1 \right \}$ are compacts or not.
I know that I should show that they admit a convergent subsequence. Any help?
A metric space with an infinite closed discrete subspace is not compact. Let $v_k=(x_{k,n})_{n\in N}$ where $x_{k,n}$ is $1$ when $k=n$, and is $0$ when $k\ne n.$ Then $\|v_k-v_j\|=1$ when $k\ne j.$ So $\{v_k: 2\leq\ k \in N\}$ is an infinite closed discrete space of $B_1$ and of $B_2.$