Is $\ell^p \mathbb N \subset \ell^q \mathbb N$ inclusion compact ?

Question

This question just struck me, is it true that if $1 <p <q <\infty$ , is the inclusion map

$$\ell^p \mathbb N \subset \ell^q \mathbb N$$compact ?

Hölders inequality gives us that the inclusion is continuous . But for compactness it doesn't seem very direct .

Thank you for ur help .

Answer 1

No: let $e^{(n)}_k:=\delta_{nk}$. Then the sequence $\{e^{(n)}\}$ is bounded in $\ell^p$, for $1\leqslant p\leqslant \infty$. If $n_1\neq n_2$, then $$\lVert e^{n_1}-e^{n_2}\rVert_q=\begin{cases} 2^{1/q},&\mbox{if }1\leqslant q<\infty\\\ 1&\mbox{if }q=+\infty, \end{cases}$$

which proves that there is no convergent subsequence in $\ell^q$. So we actually can take $1\color{red}\leqslant p\color{red}\leqslant q\color{red}\leqslant +\infty$.

Answer 2

The sequence of standard basis vectors $(1,0,\ldots),(0,1,0,\ldots),(0,0,1,0,\ldots),\ldots$ is bounded in every $\ell^p$, but has no convergent subsequence in any of these spaces.