# Pairs $(p,q)$ such that $id: l_p\to l_q$ is bounded [duplicate]

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 StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Nov 15 '15 at 9:52
• Start by finding out when you have $l_p \subset l_q$. That is a necessary condition. Is it also sufficient? – Daniel Fischer Nov 7 '15 at 13:50
• @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! – blst Nov 7 '15 at 14:16