Compact embedding $L^p$ to $L^q$

Question

Prove that this embedding is not compact $\ell^p\subset \ell^q\subset c_0$ , where $1 \leq p < q < \infty$

I understand that this embedding is not compact, but I cannot construct an example where sequence has no convergent subsequence.

$\ell^p$ space: https://en.wikipedia.org/wiki/Lp_space

$c_0$ is the space of sequences converging to 0: $\lim_{k\mapsto \infty} x_k = 0$ and metric: $\left \| x \right \| = \max\left | x_k \right |$, $k\ge1$

Any ideas would be appreciated.

Answer 1

Consider the set $X=\{e_n:n\geq1\}$ where $e_n=(0,\ldots,0,1,0,\ldots)$ with the $1$ occurring at the $n$th spot. Clearly $X\subset \ell^p$ for any $p$ and moreover it is bounded; $\|x\|_p=1$ for all $x\in X$. However, you can see that $X$ is not relatively compact in any $\ell^p$ or $c_0$. Hence the inclusions $\ell^p\subset \ell^q$ and $\ell^q\subset c_0$ are not compact.

One should compare this failure of compact embedding to the Rellich theorem---it's the unboundedness of the underlying measure space $\mathbb N$ that plays a key role.