In my book there is this definition:
Definition: The space of all square integrable functions over a given domain $\Omega\in\mathbb{R}^n$ is known as the $L_2(\Omega)$-space. If $u \in L_2(\Omega),$ then the $L_2$-norm of $u$ (in 1-D case) is $$\|u\|_{L_2(\Omega)}=\sqrt{(u,u)}=\left(\int_{\Omega}|u(x)|^2 \ dx\right)^{1/2}.\tag1$$
Questions:
- Denote the boundary of $\Omega$ as $\partial\Omega=\Gamma$. Then, is it always true that $$\|u\|_{L_2(\Omega)} \geq \|u\|_{L_2(\Gamma)}, \tag 2$$ and how can one prove/explain it?
- How would the above look like if we, for example, dealt with $\Omega=[a,b]\times[a,b]$? How would $\Gamma$ look like then?