Why is there a sequence $u_n\in X\cap \mathcal C^2[-1,1]$ s.t. $u_n\to u$ in $W^{1,2}(-1,1)$?

Question

Let $$X=\{u\in W^{1,2}(-1,1)\mid u(-1)=u(1), \int_{-1}^1 u=0\}.$$ I proved Wirtinger inequality i.e. $$\int_{-1}^1 (u')^2\geq \pi^2\int_{-1}^1 u^2$$ for all $u\in X\cap \mathcal C^2([-1,1])$. But I want to prove it when $u\in X$ (and not $X\cap \mathcal C^2([-1,1])$). In my course, it's written that $X\cap \mathcal C^2([-1,1])$ is dense in $X$, i.e. that for all $u\in X$ there is a sequence $u_n\in X\cap \mathcal C^2([-1,1])$ s.t. $u_n\to u$ in $W^{1,2}(-1,1)$. Is it a famous result ? Can someone give me a link where such density is mentioned ?

Answer 1

Typically, density of $C^2([-1,1])$ in $W^{1,2}(-1,1)$ is shown by considering a mollification $u *\phi_n$ of $u\in W^{1,2}(-1,1)$. So, one way to show what you are asking is to revisit this proof, and actually show that $u * \phi_n\in X$ whenever $u\in X$.