Theorem: Suppose $T_h$ is a regular family of triangulations of $\Omega$ which is a convex polygonal domain, then the finite element approximation $u_h \in V_h$ satisfies $$ ||u-u_h||_1\le C h||u||_2 \le C h ||f||_0$$
A proof of the above can be found here. But I am not able to follow the proof because my text follows a different notation.
Can someone explain how the following norms are defined?
$||.||_1 $
$ ||.||_2$
$ ||.||_0$
$|.|_0$
Typically, $\|\cdot\|_k$ is used to denote the norm in the Sobolev space $H^k(\Omega) = W^{k,2}(\Omega)$ with the typical convention $H^0(\Omega) = L^2(\Omega)$ to denote the Lebesgue space.
This resolves 1.-3.
I do not know what 4. means. Typically, $|.|$ is used to denote a semi-norm, but there is no reasonable semi-norm in $L^2(\Omega)$. To give an example in $H^1(\Omega)$, a typical notation is $$|u|_1 = |u|_{H^1(\Omega)} = \left(\int_\Omega \|\nabla u(x)\|^2 \, \mathrm dx\right)^{1/2}$$