Let $C_{0}(\mathbb{R}^n)$ denote the set of continuous functions in $\mathbb{R}^n$ with compact support, $C_0^\infty(\mathbb{R}^n)$ denote the set of infinite differentiable functions in $\mathbb{R}^n$ with compact support. We define $$\mathcal{E}(u,v)=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{n+\beta}}\mathrm{d}y\mathrm{d}x$$ $$\mathcal{F}=\left\{u\in L^2(\mathbb{R}^n):\mathcal{E}(u,u)<+\infty\right\}$$ where $\beta\in(0,2)$.
It is easy to prove that $(\mathcal{E},\mathcal{F})$ is a Dirichlet form. We need to show that it is regular, i.e. $C_0(\mathbb{R}^n)\cap\mathcal{F}$ is dense in $C_0(\mathbb{R}^n)$ in the uniform metric and dense in $\mathcal{F}$ in the $\sqrt{\mathcal{E}(u,u)+\lVert u\rVert_2^2}$ metric.
I can show that $C_0^\infty(\mathbb{R}^n)\subseteq C_0(\mathbb{R}^n)\cap\mathcal{F}$ using $\beta\in(0,2)$. Hence we have $C_0(\mathbb{R}^n)\cap\mathcal{F}$ is dense in $C_0(\mathbb{R}^n)$ in the uniform metric.
But I don't know how to prove $C_0(\mathbb{R}^n)\cap\mathcal{F}$ is dense in $\mathcal{F}$ in the $\sqrt{\mathcal{E}(u,u)+\lVert u\rVert_2^2}$ metric.
Can someone give some hints or references? Thanks.
