In PDE Evans 2nd edition (pages 262-263), I am trying to understand a proof of a theorem, which states:
THEOREM 2 (Sobolev spaces as function spaces). For each $k=1,\ldots$ and $1\le p\le\infty$, the Sobolev space $W^{k,p}(U)$ is a Banach space.
My question concerns verifying the triangle inequality property, which is part of showing that $\|u\|_{W^{k,p}(U)}$ is a norm.
Next assume $u,v \in W^{k,p}(U)$. Then if $1\le p < \infty$, Minkowski's inequality implies \begin{align} \|u+v\|_{W^{k,p}(U)} &= \left(\sum_{|\alpha|\le k} \|D^\alpha u + D^\alpha v \|_{L^p(U)}^p \right)^{1/p} \\ &\le \left(\sum_{|\alpha|\le k} (\|D^\alpha u\|_{L^p(U)}+\| D^\alpha v \|_{L^p(U)})^p \right)^{1/p} \\ &\le \left(\sum_{|\alpha|\le k} \|D^\alpha u\|_{L^p(U)}^p \right)^{1/p} + \left(\sum_{|\alpha|\le k} \|D^\alpha v \|_{L^p(U)}^p \right)^{1/p} \\ &= \|u\|_{W^{k,p}(U)} + \|v\|_{W^{k,p}(U)}. \end{align}
Minkowski's inequality was used to justify the first inequality. But how can we justify the second inequality $$\left(\sum_{|\alpha|\le k} (\|D^\alpha u\|_{L^p(U)}+\| D^\alpha v \|_{L^p(U)})^p \right)^{1/p} \\ \le \left(\sum_{|\alpha|\le k} \|D^\alpha u\|_{L^p(U)}^p \right)^{1/p} + \left(\sum_{|\alpha|\le k} \|D^\alpha v \|_{L^p(U)}^p \right)^{1/p}$$
The first inequality is the triangle (Minkowski) inequality for the $L^p$ norm, the second inequality is Minkowski inequality for the counting measure, http://en.wikipedia.org/wiki/Minkowski_inequality.