Ineqaulity on \ell_p

1. Oct 19, 2009

forumfann

Could anyone prove or disprove the following inequality:
$||x||_{p}\leq||x||_{p'}$ for all $x\in\mathbb{R}^{n}$ if $p'>p\geq1$?

By the way, this is not a homework problem.

Any help on this will be highly appreciated.

2. Oct 19, 2009

Staff: Mentor

Where is the problem from?

3. Oct 19, 2009

forumfann

This a problem that I was curious about, because we know that $||x||_{m}\leq||x||_{1}$ for any positive integer $m$, and then I wondered if it is true for any $p\geq1$.

But it would be great if one can show the following:
$||x||_{p}\leq||x||_{1}$ for $p\geq1$,
so could anyone help me on this?

Last edited: Oct 19, 2009
4. Oct 19, 2009

trambolin

$$\|x\|^p_p = \sum_i{|x_i|^p} \leq \left( \sum_i{(|x_i|)} \right)^p = \|x\|_1^p$$

Though I am very dizzy right now, it should be OK where I used $a^2+b^2 \leq (a+b)^2, a,b > 0$

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook