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Let $\bar{B}$ be the closed unit ball in $\mathbb{R}^n$, $C^k\left(\bar{B}\right)$ the Banach space of all real function defined on $\bar{B}$ with continuous derivatives up to order $k$, with norm $$\Vert f \Vert = \sum_{h\le k} \Vert \partial_{i_1}\dots \partial_{i_h}f\Vert_\infty$$ Are polynomials dense in $C^k\left(\bar{B}\right)$?
Can you give me references about that subject?
Thanks a lot...

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up vote 6 down vote accepted

By Stone-Weierstrass, the polynomials are dense in $C(\overline{B})$. Do the rest by induction: if polynomials are dense in $C^{k-1}$, write a function in $C^k$ in terms of integrals of its partial derivatives, and approximate those partial derivatives by polynomials...

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Sorry for being dense, but isn't $C^k(\bar{B}) \subset C(\bar{B})$ and therefore a subset of the former that is dense in the latter automatically dense in the former? Edit: Oh sorry: $C^k$ is equipped with a different norm than C... –  Tim van Beek May 19 '11 at 11:53
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