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Let $C_c^\infty(\mathbb{R})$ be the set of $C^{\infty}$ functions with compact support in $\mathbb{R}$.

Is $C_c^{\infty}(\mathbb{R})$ with the norm $\|\varphi\|_{\infty}+\| \varphi'\|_{\infty}$ separable? And in general with $$\sum_{k=1}^N\|\varphi^{(k)}\|_\infty\,?$$

user39756
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1 Answers1

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Hint: Let $\mathcal P_{\mathbb Q}$ denote the set of polynomials with rational coefficients. For each $n\in \mathbb N,$ choose $g_n \in C^\infty_c(\mathbb R)$ such that $g_n=1$ on $[-n,n].$ Consider $\{g_np: n \in \mathbb N, p\in \mathcal P_{\mathbb Q}\}.$

zhw.
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  • Why do polynomials approximate also derivatives? Stone-Weierstrass says that, for any continuous function $f$ on $[a,b]$, $p_n\rightarrow f$ uniformly, but not $p_n'\rightarrow f'$, right? – user39756 Jun 05 '17 at 15:12
  • I think that the answer to my previous comment is here. – user39756 Jun 05 '17 at 15:25