Sobolev space - norm $H^1$ and $H^1_0$

Question

When we defined on $H^1_0$ the norm $$||v||_{H^1_0}=||v||_{L^2}+||\nabla v||_{L^2}$$ can we tell that $$||u||_{H^1_0} = ||u||_{H^1}?$$ Thank's

Answer 1

Not all function spaces need their own norm definitions. Some are defined as subspaces of a larger normed space, and inherit the norm automatically.

For example, $\ell_\infty$ is the space of all bounded sequences, $c$ is its subspace that consists of convergent sequences, and $c_0$ consists of the sequences that converge to $0$. We have $\ell_\infty$ norm, but there is no reason to speak of "$c$ norm" or "$c_0$ norm". The norm is already there, inherited from the larger space.

This is how it works with $H^1$ and $H^1_0$. We define a norm on $H^1$. Then we define its subspace $H^1_0$, which already has a norm: it gets it from $H^1$. No need to speak of "$H^1_0$ norm".

That said, on certain domains one can prove that for $u\in H^1_0$, the $H^1$ norm is equivalent to $\|\nabla u\|_{L^2}$ (the homogeneous $H^1$ seminorm), and use $\|\nabla u\|_{L^2}$ as a norm on $H^1_0$.