# Question on norm on Sobolev Space

If $$u\in W^{k,p}(\Omega)$$ we define its norm as $$\Vert u \Vert_{W^{k,p}(\Omega)}= \begin{cases} \displaystyle \left(\sum_{|\alpha|\le k}\displaystyle\int\limits_\Omega |D^\alpha u|^p\mathrm{d}x\right)^\frac{1}{p} & 1\le p <\infty,\\ \\ \:\:\displaystyle\sum_{|\alpha|\le k} \underset{\Omega}{\mathrm{ess\,sup}}|D^\alpha u| & p=\infty \end{cases}$$

Are these the only norms on a Sobolev space... and how can we say that these norms exists on this space?

There are other equivalents norm as for example $$\|u\|_{W^{k,p}}=\sum_{|\alpha |\leq k}\|D^\alpha u\|_{L^p}$$ or $$\|u\|_{W^{k,p}}^p=\sum_{|\alpha |\leq k}\|D^\alpha u\|_{L^p}^p.$$
For the existence, what do you mean ? $$\|\cdot \|_{W^{k,p}}$$ is a norm on $$W^{k,p}$$, so it exist !
• @InverseProblem: What do you mean by "well defined" ? That $\|u\|_{W^{k,p}}<\infty$ ? If yes, then it's well defined on $W^{k,p}$ by definition of $W^{k,p}$. – Surb Feb 9 at 14:25