Separability of continuous functions with compact support - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-24T00:44:48Z http://mathoverflow.net/feeds/question/67763 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67763/separability-of-continuous-functions-with-compact-support Separability of continuous functions with compact support fjodor_d 2011-06-14T14:06:34Z 2011-06-14T14:25:47Z <p>Hi,</p> <p>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?</p> <p>Please note: this is not a duplicate of this question <a href="http://mathoverflow.net/questions/54026/separability-of-a-certain-space-of-continuous-functions" rel="nofollow">(click)</a>, since I asked for compact support.</p> <p>The reason I'm asking is: here <a href="http://www.google.com/url?sa=t&amp;source=web&amp;cd=1&amp;ved=0CCMQFjAA&amp;url=http%3A%2F%2Fdiffusion.jlu.edu.cn%2Fseminar%2Ffiles%2FEvans.pdf&amp;rct=j&amp;q=evans%20weak%20convergence%20methods%20for%20nonlinear%20partial%20differential%20equations&amp;ei=BGj3Tav2N5C_gQf8zpGTDA&amp;usg=AFQjCNF3kplJO_opo1rtuN_WhvNHkaYeHg&amp;cad=rja" rel="nofollow">(click)</a> 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.</p> <p>Your help is very much appreciated! Thanks!</p> http://mathoverflow.net/questions/67763/separability-of-continuous-functions-with-compact-support/67769#67769 Answer by unknown (google) for Separability of continuous functions with compact support unknown (google) 2011-06-14T14:25:47Z 2011-06-14T14:25:47Z <p>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 $C_0(\mathbb R^m)$.</p>