
Embeddings of norms
Just in case, here is a definition that I'm working with
Let $\displaystyle X$ be a linear space, with the norm $\displaystyle \cdot_X$, and $\displaystyle Y$ be a linear space with the norm $\displaystyle \cdot_Y$. If $\displaystyle X$ is a subspace of $\displaystyle Y$ and there exists a constant $\displaystyle C$ such that for all $\displaystyle f \in X$, one has $\displaystyle f_Y \leq Cf_X$, then we write $\displaystyle X \subset Y$, and we say that there is an embedding of $\displaystyle X$ into $\displaystyle Y$.
Now, I want to prove that if $\displaystyle p < r$, then $\displaystyle l^p \subset l^{\infty}$. The hint to this question says the following
"First prove that $\displaystyle l^p \subset l^{\infty}$. Then write $\displaystyle r=p+s, s>0$ and use the inequality $\displaystyle x_i^r \leq x_i^px_{\infty}^s$"
I don't think I've understood the hint properly, because somehow I'm not using the first half of the hint. I've found that the second part of the hint is sufficient, here's what I've done so far;
We know that $\displaystyle x_i^s \leq (\sup_i x_i)^s = x_i_{\infty}^s$. Now, we know that $\displaystyle r=s+p$, and so we have $\displaystyle x_i^r \leq x_i^px_{\infty}^s$. Then,
$\displaystyle \sum_i x_i^r \leq \sum_i x_i^px_{\infty}^s = x_{\infty}^s\sum_i x_i^p$. Now, given that $\displaystyle r > p$, then $\displaystyle \frac{1}{r} < \frac{1}{p}$,
and so
$\displaystyle \left(\sum_i x_i^r\right)^{1/r} \leq x_{\infty}^{s/p}\left(\sum_i x_i^p\right)^{1/p}$
which is just $\displaystyle x_r \leq x_{\infty}^{s/p}x_p$. Comparing this to the definition, we find that $\displaystyle l^p \subset l^r$.
Have I missed something here, I'm not entirely sure if this is correct.
Thanks a lot in advance,
HTale.