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I'm trying to show the inclusion :

$\ell^p\subseteq\ell^q$ for real-value sequences, and show that the norms satisfy: $\|\cdot\|_q<\|\cdot\|_p$.

I think I can show the first part without much trouble:

Take $a_n$ in $\ell^p$, then the partial sums are a Cauchy sequence, i.e., for any $\epsilon>0$ , there is a natural $N$ with $|S_{n,p}-S_{k,p}|<\epsilon$ for $n,k>N$, and $S_{n,p}$ the partial sums of $|a_n|^p$ and the individual terms go to $0$. So, we choose an index $J$ with $a_j<1$ for $j>J$. We then use that $f(x)=a^x$ decreases in $[0,1]$. This means that $|a_j|^p<|a_j|^q$.

So the tail of $S_{n,q}$, the partial sums of $|a_n|^q$ decrease fast-enough to converge, by comparison with the tail of $S_{n,p}$.

But I'm having trouble showing $\|\cdot\|_q<\|\cdot\|_p$ . Also, is there a specific canonical embedding between the two spaces?

  • 3
    A similar question is [here](http://math.stackexchange.com/questions/4094/how-do-you-show-that-l-p-subset-l-q-for-p-leq-q)2012-02-29
  • 0
    I think you want $\Vert\ \cdot\ \Vert_q\le \Vert\ \cdot\ \Vert_p$.2012-02-29
  • 0
    Right, thanks for the ref., let me rewrite.2012-02-29

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