Why is $D(\Omega)$ continuously embeded in $H_0^1(\Omega)$? let $D(\Omega)$ is the space of $C^{\infty}$ functions with compact support, with the usual notion of convergence we give in the context of distributions.
In my notes, before showing that $H^{-1} \subset D'(\Omega)$, my professor wrote that

$D(\Omega) \hookrightarrow H_0^1(\Omega)$ is a continuous embedding

I can't understand why this is true. If I consider the immersion $\mathcal{i}: D(\Omega) \rightarrow H_0^1(\Omega)$ with $i(v)=v$, I should be able to prove $$||i(v)||_{H_0^1}=||v||_{H_0^1} \leq ||v||_{D(\Omega)}$$ but I do not know what is the norm on $D(\Omega)$! I am missing some fundamental thing here, and indeed in my book I can't find what is the norm given on $D(\Omega)$. What am I missing?
 A: As per Exercise 3 in the answer I linked to
Doubt in understanding Space $D(\Omega)$
what one must show is that for every continuous seminorm $\eta$ on $H_0^1(\Omega)$, there exists a continuous seminorm $\rho$ on $\mathscr{D}(\Omega)$ such that, for all $v\in\mathscr{D}(\Omega)$,
$$
\eta(i(v))\le \rho(v)\ .
$$
Now it is enough to do this $\eta$'s which define the locally convex topology of $H_0^1(\Omega)$, namely just $\eta=\|\cdot\|_{H_0^1}$.
To construct $\rho$, just take $\rho(v):=\|i(v)\|_{H_0^1}$. To show it is a continuous seminorm, it is enough to show it is an admissible seminorm on $\mathscr{D}(\Omega)$, i.e., belongs to the set $B$ given in Example 2 of the post I mentioned.
More concretely, what you need to show is that for all compact $K\subset\Omega$, there exists $C\ge 0$ and $N\ge 0$, such that for all $v\in\mathscr{D}(\Omega)$ with support contained in $K$, we have
$$
\|v\|_{H_0^1}\le C\sup_{x\in K}\max_{|\alpha|\le N} |D^{\alpha}v(x)|
$$
which a multivariate calculus exercise.
