I am having a bit of trouble understanding the Sobolev space and the Sobolev norm.
Am I correct in saying that, for example, in the space $W^{1,2}(0,1)$ we have
$$\lVert f\rVert_{W^{1,2}(\Omega)}=\bigg[\int_0^1\bigg(\bigg(\frac{df}{dx}\bigg)^2+f^2 \bigg)\, dx \bigg]^{1/2}$$
and that
$$W^{1,2}(0,1)=\big\{f \in L^1_{loc}(0,1):\lVert f\rVert_{W^{1,2}(\Omega)} \leq \infty \big\}$$