Checking the Sobolev norm

Question

I am trying to show that the Sobolev norm is indeed a norm. Why is the following statement true?

For $p \in [1, \infty)$, $$ \left(\sum_{|\alpha|\le k}(||D^{\alpha}u||_{L^p(U)} + ||D^{\alpha}v||_{L^p(U)})^p\right)^{\frac{1}{p}} \le \left(\sum_{|\alpha|\le k}||D^{\alpha}u||_{L^p(U)}^p\right)^{\frac{1}{p}} + \left(\sum_{|\alpha|\le k}||D^{\alpha}v||_{L^p(U)}^p\right)^{\frac{1}{p}} $$

Answer 1

This is Minkowski's Inequality for sums.