How can we show that $L^2$ is densely embedded into $H^{-1}$? Let


*

*$d\in\mathbb N$

*$\Lambda\subseteq\mathbb R^d$ be open

*$H^{-1}(\Lambda):=H_0^1(\Lambda)'$ denote the topological dual sapce of $H_0^1(\Lambda)$


Since $$\langle u,f\rangle_{H_0^1,\:H^{-1}}:=\langle u,f\rangle_{L^2(\Lambda)}\;\;\;\text{for }u\in H_0^1(\Lambda)$$ is a bounded linear functional on $H_0^1(\Lambda)$ for all $f\in L^2(\Lambda)$, $$\iota:L^2(\Lambda)\to H^{-1}(\Lambda)\;,\;\;\;f\mapsto\langle\;\cdot\;,f\rangle_{H_0^1,\:H^{-1}}$$ is well-defined. Note that $\iota$ is injective and linear.

How can we show that $\iota L^2(\Lambda)$ is a dense in $H^{-1}(\Lambda)$?

I guess we somehow need to use the Hahn-Banach theorem or the Stone–Weierstrass theorem. For the first one, we would need to show that $\left.\eta\right|_{\iota L^2(\Lambda)}=0$ implies $\eta=0$, for all $\eta\in H^{-1}(\Lambda)'$.
 A: Let $F \in H^{-1}$.  If $\Lambda$ is bounded take $\kappa \ge 0$ and if $\Lambda$ is unbounded take $\kappa >0$.  We will take the inner-product on $H^1_0$ to be 
$$
(u,v)_{H^1_0} = \int_\Lambda\left( \nabla u \cdot \nabla v + \kappa uv\right).
$$
Using Riesz / the weak-solvability of $-\Delta u +\kappa u =F$ with Dirichlet BCs, we can find a unique $u \in H^1_0$ such that 
$$
\langle F,v \rangle = \int_\Lambda \left(\nabla u \cdot \nabla v + \kappa uv \right) \text{ for all }v \in H^1_0.
$$
Let $\epsilon >0$.  Since $C^\infty_c(\Lambda)$ is dense in $H^1_0$ we can find $u_\epsilon \in C^\infty_c$ such that 
$$
\Vert u - u_\epsilon \Vert_{H^1_0} = \left(  \int_\Lambda |\nabla u - \nabla u_\epsilon|^2 + \kappa|u-u_\epsilon|^2\right)^{1/2} < \epsilon.
$$ 
Then 
$$
\langle F,v \rangle = \int_{\Lambda}\left[ (\nabla u -\nabla u_\epsilon) \cdot \nabla v + \kappa (u-u_\epsilon)v\right] + \int_\Lambda \left(\nabla u_\epsilon \cdot \nabla v + \kappa u_\epsilon v \right)
$$
for $v \in H^1_0$.  Since $u_\epsilon \in C^\infty_c$ we can rewrite the last term as
$$
\int_\Lambda \left(\nabla u_\epsilon \cdot \nabla v + \kappa u_\epsilon v\right)= \int_\Lambda (- \Delta u_\epsilon + \kappa u_\epsilon) v = (-\Delta u_\epsilon+\kappa u_\epsilon,v)_{L^2} = \langle \iota (-\Delta u_\epsilon+ \kappa u_\epsilon), v\rangle.
$$
Thus 
$$
\langle F - \iota(-\Delta u_\epsilon + \kappa u_\epsilon),v\rangle = \int_{\Lambda}\left[ (\nabla u -\nabla u_\epsilon) \cdot \nabla v + \kappa (u-u_\epsilon)v\right]
$$
and we can estimate
$$
\Vert F - \iota(-\Delta u_\epsilon+ \kappa u_\epsilon)\Vert_{H^{-1}} = \sup_{\Vert v \Vert_{H^1_0} \le 1}  \langle F - \iota(-\Delta u_\epsilon+ \kappa u_\epsilon),v\rangle \le \Vert u - u_\epsilon \Vert_{H^1_0} < \epsilon.
$$
Since $\epsilon >0$ was arbitrary we deduce that $\iota L^2$ is dense in $H^{-1}$.
