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$U$ 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 $C^k([0,1])$?

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  • $\begingroup$ Can you define $C_0^\infty(U)$? And explain how to view $C_0^\infty((0,1))$ as a subspace of $C^k([0,1])$? $\endgroup$ – jathd Dec 2 '12 at 9:24
  • $\begingroup$ 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. $\endgroup$ – Jay Dec 2 '12 at 9:50
  • $\begingroup$ What are the involved topologies? $\endgroup$ – Davide Giraudo Dec 2 '12 at 13:48
  • $\begingroup$ @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. $\endgroup$ – Jay Dec 2 '12 at 15:56
  • $\begingroup$ 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 $C^k(U)$? $\endgroup$ – Lukas Geyer Dec 4 '12 at 1:42

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