# triangle inequality

• Jan 28th 2009, 07:43 AM
asi123
triangle inequality
Hey guys.
How can I prove the triangle inequality for this norm.
I marked in green what I did but then I got stuck.

• Jan 28th 2009, 10:23 AM
Opalg
This is Minkowski's inequality. You can find proofs online if you Google for them.

\displaystyle \begin{aligned}\|x+y\|_p^p = \sum|x_i+y_i|^p &\leqslant \sum\bigl(|x_i|+|y_i|\bigr)|x_i+y_i|^{p-1} \\ &= \sum\bigl|x_i||x_i+y_i|^{p-1} + \sum|y_i||x_i+y_i|^{p-1},\end{aligned}
and then use Hölder's inequality on each of those two sums.
• Jan 28th 2009, 10:24 AM
Moo
Hello,
Quote:

Originally Posted by asi123
Hey guys.
How can I prove the triangle inequality for this norm.
I marked in green what I did but then I got stuck.

Consider $\displaystyle \|x+y\|_p^p=\sum_{i=1}^\infty |x_i+y_i|^p$
And then you know that $\displaystyle |m+n| \leqslant |m|+|n|$
So $\displaystyle \forall i \geqslant 1,~ |x_i+y_i|\leqslant |x_i|+|y_i|$, which implies $\displaystyle |x_i+y_i|^p \leqslant (|x_i|+|y_i|)^p$

And hence $\displaystyle \|x+y\|_p^p=\sum_{i=1}^\infty |x_i+y_i|^p \leqslant \sum_{i=1}^\infty (|x_i|+|y_i|)^p$

Now take the p-th root and you're done.
• Jan 28th 2009, 10:29 AM
asi123
Quote:

Originally Posted by Moo
take the p-th root and you're done.

Do what?

Thanks a lot.
• Jan 28th 2009, 10:34 AM
Moo
Quote:

Originally Posted by asi123
Do what?

Thanks a lot.

Nah >< I'm really sorry, I thought you wanted to prove the green stuff (Worried)

(p-th root of x is $\displaystyle x^{1/p}$)