In my finite element notes, after the proof of the global estimate for the interpolation error, assuming a regular triangulation with triangles $T_m$:
$$\sum_m|v - \Pi_h^r v|_{s,p,T_m} \leq \sigma^{-s} h^{k+1-s}|u|_{k+1,\Omega}$$ for $u \in W^{k+1,p}(\Omega)$
After that, the professor told that in general $$\sum_m |v - \Pi_h^r v|_{k,p,T_m}^p \ne |v - \Pi_h^r v|_{k,p,\Omega}^p$$ I'm trying to find an example of this All I know is that, for instance, the space $X_h =\{ v \in C^o (\Omega): v_{|T_m} \in P^1(T_m) \}$ is not a subspace of $H^2(\Omega)$ since the second derivative has delta distributions.
I'd like to find even 1D example for which the sum over the intervals of the norms is not equal to the norm on the whole space, but I can't come up with anything.
I guess that the problem really is $v - \Pi_h^r v \not\in W^{k,p}(\Omega)$.
To give an example, consider $\Omega = (-1,1)$ with the triangulation $T_0 = (-1,0)$, $T_2 = (0,1)$. Then, the function $w$ defined via $w(x) = |x|$ belongs to a $P^1$ space and $w \not\in H^2(\Omega)$. In particular, $$ \sum_{i = 1}^2 | w |_{2,2,T_i}^2 = 0 \ne \infty = | w |_{2,2,\Omega}^2. $$