Continuously differentiable functions is closed in $H^1(\Omega)$

Question

Let $\Omega \subset \mathbb{R}^N$ a bounded smooth domain. I would like to know if the space $C^{1}(\overline{\Omega})$ is a closed subspace of $H^1(\Omega)$ in the $H^1(\Omega)$ norm. I'm trying to prove this, because I need to introduce an inner product on $C^{1}(\overline{\Omega})$ to make it a Hilbert space.

Answer 1

This is not true. Take $\Omega=(-1,1)$, $y(x)=|x|$. Define $y_k(x):=\sqrt{y(x)^2 + \frac1k}$. Then $y_k \to y$ in $H^1$, which can be proven by dominated convergence. And $C^1$ is not a closed subspace in $H^1$.