Hi,


is the space $C\_0(\mathbb{R}^m)$, $m \in \mathbb{N}$ of continuous functions with compact support separable? If yes: where can I find a proof for that?


Please note: this is not a duplicate of this question [(click)](https://mathoverflow.net/questions/54026/separability-of-a-certain-space-of-continuous-functions), since I asked for compact support.


The reason I'm asking is: here [(click)](http://www.google.com/url?sa=t&source=web&cd=1&ved=0CCMQFjAA&url=http%3A%2F%2Fdiffusion.jlu.edu.cn%2Fseminar%2Ffiles%2FEvans.pdf&rct=j&q=evans%20weak%20convergence%20methods%20for%20nonlinear%20partial%20differential%20equations&ei=BGj3Tav2N5C_gQf8zpGTDA&usg=AFQjCNF3kplJO_opo1rtuN_WhvNHkaYeHg&cad=rja) the author (L.C. Evans) seems to use the fact on page 18 (page 20 of the pdf) first line without mentioning the difficulty, that $\mathbb{R}$ isn't compact. The entire proof is senseless without this fact.


Your help is very much appreciated!
Thanks!



Separability of continuous functions with compact support
Each $C([-N,N]^m)$ is separable, and so is the subspace $X\_N\subseteq C([-N,N]^m)$ consisting of functions which vanish on the boundary $\partial[-N,N]^m$. Take a countable dense subset of each $X\_N$, and take the union over all integers $N$. This is then a countable dense subset of $C\_0(\mathbb R^m)$.


