Inclusion of $L^p$ spaces for functions has been discussed here.
Does this apply to $l^p$ space of sequences similarly?
I tried to show the following: For $1\leq p<q<\infty$, $l^q\subset l^p$ By using Hölder inequality but it doesn't seem to work.
My question is that is this true? If yes, what's the right way to prove it and what's a good counter example for showing $l^p\subset l^q$ is not true? Thanks.
If $\sum_n |x_n|^p < \infty$, then $|x_n| \leq 1$ definitely. Therefore, if $q>p$, then $|x_n|^q \leq |x_n|^p$, and we conclude that $\ell^p \subset \ell^q$.
Now the opposite embedding can't be true, otherwise $\ell^p \simeq \ell^q$ for every pair $(p,q)$, and this is obviously false.