# $C_c^{\infty}(\Omega)$ dense in $L^2(\mathbb{\Omega})$

Let $\Omega:= ]0,2\pi[^d\subset \mathbb R^d$, and let's consider following spaces:

• $L^2(\mathbb{\Omega}) = L^2(\Omega, \mu_L)$ := space of $\Omega$-valued functions defined on $\Omega$ for which the square of the absolute value is Lebesgue integrable relatively to standart Lebesgue measure.

• $C_c^{\infty}(\Omega) = \{f\in C^{\infty}(\Omega):\,\,\overline{\mathrm{supp}\,f}\subsetneq \Omega\}$ := space of functions that are both smooth, in the sense of having continuous (strong) derivatives of all orders, and compactly supported.

I'm looking for the proof of : $C_c^{\infty}(\Omega)$ is dense in $L^2(\Omega)$.

• Hint: Use the fact that simple functions are dense in $L^2$ Jun 5, 2017 at 10:54

Let $$f\in L^2(]0, 2\pi[)$$ and take an approximation to identity $$\eta_\epsilon \in C_c^\infty(\mathbb R)$$ such that $$\operatorname{supp} \eta_\epsilon \subset ]-\epsilon, \epsilon[.$$ Then let $$f_\epsilon = \eta_\epsilon * f \chi_{[2\epsilon, 2\pi-2\epsilon]}$$ where $$\chi_A(x) = 1$$ if $$x\in A$$ and $$=0$$ if $$x \notin A$$. Then $$f_\epsilon \in C_c^\infty(\Omega)$$ with support $$\subset [\epsilon,2\pi-\epsilon]$$ and $$f_\epsilon \to f$$ in $$L^2(\Omega)$$.