2

Is $C^{\infty}_c (R^d)$ dense in $C_c(R^d)$?

with $C_c(R^d)$ the space of continuous functions $R^d \rightarrow R$, that have compact support.

If yes, do you have a proof?

Thanks

mimi
  • 853

1 Answers1

2

Of course it is.

You can prove it by convoluting $C_c(\mathbb{R}^d)$ functions with a smooth and compactly supported approximation of unity (also called mollifier). This gives a smooth uniform approximation of the original function.

The same argument shows that $C_c(\mathbb{R}^d)$ is dense in $L^p(\mathbb{R}^d)$ for $1\le p<\infty$.

J.R.
  • 17,904
  • Some details this answer: http://math.stackexchange.com/questions/242877/compact-support-functions-dense-in-l-1/242895?noredirect=1#comment1402033_242895 – Martin Argerami Feb 13 '14 at 15:20