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.

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    $\begingroup$ If a sequence $\{x_n\}_n$ is in $\ell^p$, then $x_n \to 0$. This implies a natural embedding of $\ell^p$ into $\ell^q$. $\endgroup$ – Siminore Feb 9 '14 at 9:13
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    $\begingroup$ See this. $\endgroup$ – David Mitra Feb 9 '14 at 9:33
  • $\begingroup$ Thanks. so should I delete my question? $\endgroup$ – Spock Feb 9 '14 at 9:42

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.

  • $\begingroup$ $|x_n|^q \leq |x_n|^p$ is true. but how do we control their powers since we have $(\sum_n |x_n|^p)^{1/p}$? $\endgroup$ – Spock Feb 9 '14 at 9:20
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    $\begingroup$ Also, how is $|x_n| \leq 1$ if we have a sequence with $2$ in the first term and infinite $0$s? $\endgroup$ – Spock Feb 9 '14 at 9:29
  • $\begingroup$ Come on, you can do better that this! I wrote definitely, i.e. beyond some large index. You should not care of the first terms, since they can be always bounded by some fixed constant. Moreover, $|x_n|^q \leq |x_n|^p$ implies that $\|x_n\|_q^q \leq \|x_n\|_p^p$ by summation. Hence the injection $\imath \colon \ell^p \to \ell^q$ is continuous. $\endgroup$ – Siminore Feb 9 '14 at 13:29
  • $\begingroup$ "can't be true because it's obviously false" does not make a good argument. You also commit a logical fallacy saying "for every pair". Maybe it's true for some pairs, not for all of them. $\endgroup$ – user147263 Nov 15 '15 at 7:45

For completeness, a counterexample for $p>q$: pick $s\in (1/p,1/q)$ and consider the sequence $x_n=1/n^s$. This sequence is in $\ell^p$ but not in $\ell^q$.


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