Proving compactness of an operator

Question

I'm having a hard time finding a solution for the following problem:

Prove that the operator $ T \in \mathcal{L}(\ell_2) $ defined with the formula $$ T((x_1, x_2, \dots, )) = (0, x_1, x_2/2, x_3/3, \dots) $$

is compact.

That's the trouble I'm having with functional analysis - how do I ever know how to approach a problem like this? I'm supposed to show that $T$ turns bounded sets into relatively compact sets (it's enough that it does so for a unit ball). However I can't seem to find a way to solve that

Answer 1

Let $T_n(x) = \sum_{k=1}^n {1 \over k} x_k e_{k+1}$. $T_n$ has finite rank, so it is compact.

We have $\|T-T_n\| \le {1 \over n+1}$, hence $T_n \to T$ and so $T$ is compact.

Answer 2

Hint: You might show the image of the unit ball is totally bounded. Given $\epsilon > 0$, take $N$ large enough, and cover the projection of the image of the unit ball onto the first $N$ coordinates by small balls ...