I'm trying to solve an exercise in Brezis' Functional Analysis, i.e.,
Let $p \in [1, \infty)$. Consider $(\ell^p, |\cdot|_p)$ and $(c_0, |\cdot|_\infty)$. Check that $\ell^p \subset c_0$ with continuous injection. Is this injection compact?
There are possibly subtle mistakes that I could not recognize in below attempt. Could you have a check on it? Thank you so much for your help!
Let $J:\ell^p \to c_0, (x_n) \mapsto (x_n)$. Then $$ \|J\| = \sup_{(x_n) \in \ell^p} \frac{\sup_n |x_n|}{\left ( \sum_n |x_n|^p \right )^{1/p}} \le \sup_{(x_n) \in \ell^p} \frac{\sup_n |x_n|}{\left ( \sup_n |x_n|^p \right )^{1/p}} =1. $$
Then $J$ is linear continuous. For $n \ge 1$, we define $e_n :=(e_{n,m})_m \in \ell^p$ by $e_{n,m} = 1$ if $n=m$ and $0$ otherwise. Then $|e_n|_p =1$ for $n\ge 1$. Also, $|Je_n-Je_m|_\infty = |e_n-e_m|_\infty = 1$ for $n \neq m$. Then $(Je_n)_n$ does not have any convergent subsequence. So $J$ is not compact.