The definition of broken Sobolev space is: Let $\Omega$ be the domain and $\mathcal{E}_h$ be the finite number of partitions of $\Omega$. For any real number $s \geq 0$, we denote the broken Sobolev space \begin{equation} H^s\left(\mathcal{E}_h\right)=\left\{\mathsf{u} \in L^2(\Omega): ~\text{for all}~ E \in \mathcal{E}_h,~\left.\mathsf{u}\right|_E \in H^s(E)\right\}, \end{equation} equipped with the broken Sobolev norm: \begin{equation} \|\mathsf{u}\|\strut_{H^s\left(\mathcal{E}_h\right)}=\left(\sum_{E \in \mathcal{E}_h}\|\mathsf{u}\|\strut_{H^s(E)}^2\right)^{1/2}. \end{equation} Here, $\|\mathsf{u}\|\strut_{H^s(E)}^2$ is a Sobolev norm on an element E of the domain $\Omega$.
Please give some ideas with an explanation, how I can prove or disprove.