l2 compact operator

Question

Definined the mapping T: -> so that: where I'd like to show that if T is a compact operator then I thought i could negate my thesis and then show that T would not be compact but i wasn't able to show it. Has anyone an idea? Thanks!

Answer 1

Hint:

If $\alpha$ is not in $c_0$, then you can find a number $\varepsilon > 0$ and a subsequence $(\alpha(n_k))_{k \in \mathbb{N}}$ such that $|\alpha(n_k)| \ge \varepsilon$ for each $k \in \mathbb{N}$.

Use this to prove that the sequence $(Te_{n_k})_{k \in \mathbb{N}}$ in $\ell^2$ has no convergent subsequence (where $e_n \in \ell^2$ denotes the $n$-the canonical unit vector).