TITLE: Why do we need semi-norms on Sobolev-spaces?
QUESTION [20 upvotes]: I have been studying Sobolev spaces and easy PDEs on those spaces for a while now and  keep wondering about the norms on these spaces.
We obviously have the usual norm $\|\cdot\|_{W^{k,p}}$, but some proofs also use the semi-norm $|\cdot|_{W^{k,p}}$ and I don't really understand why we make use of this semi-norm? Why is the full norm $\|\cdot\|_{W^{k,p}}$ not enough?
I started arguing that the semi-norm is easier to calculate (in case one really wishes to do that), but I don't think that really is the reason, after all, I have never seen anyone really calculate these norms for proofs or anything. So are there other reasons for the use of this semi-norm?

REPLY [3 votes]: Really nice answer! One more important reason to use the seminorm: if you are trying to solve elliptic partial differential equations in unbounded domains, say, $$\Delta u=f\quad\text{in }\mathbb{R}^N,$$
where, say, $f\in L^p(\mathbb{R}^N)$, $1<p<N/2$, then using Green functions and singular integrals you can show that the second order derivatives of $u$ will be in $L^p(\mathbb{R}^N)$ but the function $u$ or its first derivatives will not be in $L^p(\mathbb{R}^N)$ in general, so $u$ will not belong to the space $W^{2.p}(\mathbb{R}^N)$.