# Density problem

$U$ is any open set of $\mathbb{R}$. We known that $C_0^\infty(U)$ is dense in $C^k(U)$. But what about, say $C_0^\infty((0,1))$ in $C^k([0,1])$?

• Can you define $C_0^\infty(U)$? And explain how to view $C_0^\infty((0,1))$ as a subspace of $C^k([0,1])$? – jathd Dec 2 '12 at 9:24
• It is defined as the set of complex-valued functions infinitely differentiable and whose support is included in a compact set. $k$ is a non-negative integer. – Jay Dec 2 '12 at 9:50
• What are the involved topologies? – Davide Giraudo Dec 2 '12 at 13:48
• @Davide: $C_0^\infty((0,1))$ is seen as a LF-space, $C^k$ a Banach or Fréchet space depending on the interval is open or closed. – Jay Dec 2 '12 at 15:56
• How is $C_0^\infty(U)$ dense in $C^k(U)$? E.g., how do you approximate the constant function $f\equiv 1$ in $U=(0,1)$ with $C_0^\infty((0,1))$ functions in $C^k(U)$? – Lukas Geyer Dec 4 '12 at 1:42