Why is $L^2_0$ a complete subspace?

Question

Let $\Omega \subset \mathbb{R}^N$ be an open and bounded domain. How to reason that: $$L^2_0(\Omega) = \{ q \in L^2(\Omega)\, |\, \int_{\Omega} q = 0\},$$ is a complete space?

Answer 1

Note that $\int_\Omega q$ is the inner product $\langle q , \mathbf1_\Omega\rangle$, where $\mathbf1_\Omega \in L^2$, so the map $q \mapsto \int_\Omega q$ is continuous (by Hölder's inequality).