Hey guys.
How can I prove the triangle inequality for this norm.
I marked in green what I did but then I got stuck.
Thanks in advance.
This is Minkowski's inequality. You can find proofs online if you Google for them.
The basic idea is start with
$\displaystyle \begin{aligned}\|x+y\|_p^p = \sum|x_i+y_i|^p &\leqslant \sum\bigl(|x_i|+|y_i|\bigr)|x_i+y_i|^{p-1} \\ &= \sum\bigl|x_i||x_i+y_i|^{p-1} + \sum|y_i||x_i+y_i|^{p-1},\end{aligned}$
and then use Hölder's inequality on each of those two sums.
Hello,
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.