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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 ....

a.p
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    The problem with $c_{0}$ in its usual norm is things can "run off to infinity" (i.e. pathological behavior can happen at coordinates with higher and higher indices). Try to find a norm on $c_{0}$ that discounts that behavior. –  Jan 04 '21 at 05:04

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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.

Martin Argerami
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