Show that the space of the harmonic functions over $\Omega$ in $L^2$ are closed.

Question

Let $\Omega\subset \mathbb{R}^N$ a open set and $H(\Omega)$ the space of the harmonic functions over $\Omega$ that are in $L^2(\Omega)$. Show that $H(\Omega)$ is a closed subset of $L^2$.

Answer 1

The elements of $H(\Omega)$ are the elements $u\in\mathbb L^2(\Omega)$ which satisfy the equation $$\int_{\Omega}u(x)\Delta \varphi(x)\mathrm dx=0$$ for any $\varphi$ smooth with compact support. It's the orthogonal of the subspace $V:=\{\Delta \varphi,\varphi\mbox{ smooth with compact support}\}$.