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find all pairs $p,q\in [1,\infty)$ such that $id: l_p\to l_q$ is bounded.

This just means I must find all $(p,q)$ such that $\|x\|_q \le C\|x\|_p$ for some $C$ dependent on $p$, $q$. I don't know where to start, it seems like I'm missing something obvious


marked as duplicate by user147263, Strants, Najib Idrissi, Davide Giraudo functional-analysis Nov 15 '15 at 9:52

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    $\begingroup$ Start by finding out when you have $l_p \subset l_q$. That is a necessary condition. Is it also sufficient? $\endgroup$ – Daniel Fischer Nov 7 '15 at 13:50
  • $\begingroup$ @DanielFischer We have $l_p\subset l_q$ iff p<q. It is also sufficient, take for example $y_k=\dfrac{1}{k^(1/p)}$. It is in $l_q$ but not in $l_p$. Thanks! $\endgroup$ – blst Nov 7 '15 at 14:16
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    $\begingroup$ You can have a look at this question and the posts linked there. For a more general version, see this question and other posts linked there. $\endgroup$ – Martin Sleziak Nov 7 '15 at 14:28