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 $C^k\left(\bar{B}\right)$?
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Are polynomials dense in $C^k\left(\bar{B}\right)$?
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functional-analysis
1 Answers
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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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0The integrands in the definition of $p$ should be $p_1$ and $p_2$. – 2018-06-25