TITLE: Compact embedding of $l^p$ to $c_0$
QUESTION [5 upvotes]: if $c_0$ is a space of real sequences that converges to zero with sup norm. I can show that $l^{p} \hookrightarrow c_{0}$ embedding is not compact with the sup norm. But I want something more intresting than that. I want to find norm on $c_0$ such that this emdedding be compact! But until now I can't find such norm. So if one have any idea about this ....

REPLY [3 votes]: Let $b=\{b_k\}\subset\ell^q$ with $\|b\|_q=1$ and $b_k\geq0$ for all $k$. Define
$$
\|x\|_c=\sum_{k=1}^\infty b_k\,|x_k|
$$
and $$T_nx=\sum_{k=1}^n x_ne_n,$$ where $\{e_n\}$ denotes the canonical basis in $c_0$. Then
$$
\|x-T_nx\|_c=\sum_{k>n}b_k\,|x_k|\leq\Big(\sum_{k>n}|b_k|^q\Big)^{1/q}\,\|x\|_p.
$$
Then
$$
\|I-T_n\|\leq\Big(\sum_{k>n}|b_k|^q\Big)^{1/q}\xrightarrow[n\to\infty]{}0.
$$
Thus $I$ is a limit of finite-rank operators and thus compact.