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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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    Can you define $C_0^\infty(U)$? And explain how to view $C_0^\infty((0,1))$ as a subspace of $C^k([0,1])$?2012-12-02
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    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.2012-12-02
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    What are the involved topologies?2012-12-02
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    @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.2012-12-02
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    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 $C^k(U)$?2012-12-04

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