$U$ 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 $C^k([0,1])$?
Density problem
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analysis
topological-vector-spaces
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0How 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