Let us define $C_0$ as the space of all $f \in C^0(\mathbb{R}^n,\mathbb{R})$ such that $\lim_{\|x\| \to +\infty} \|f(x)\|=0$.
Let be $\sigma>0$ fixed and define $\mathscr{H}$ the vector space generated by all function, for $v \in \mathbb{R}^n$: $$g_v(u)=\frac{1}{(2\pi \sigma^2)^{\frac n 2}}e^{-\frac{\|u-v\|^2}{2\sigma^2}}$$ for all $u \in \mathbb{R}^n$ where $\|\cdot\|$ denotes the euclidean norm in $\mathbb{R}^n$.
How can I prove that $\mathscr{H}$ is dense in $C_0$?
I know that is related with the Stone-Weiestrass theorem but I don't know how.