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Let $V$ be a Banach space and $U$ an open subset of $\mathbb{R}^n$. Let $C^\infty(U, V)$ denote the Fréchet space of $V$-valued smooth functions on $U$ with seminorms $$ \| u\|_{\alpha,K} = \sup_{y \in K} | D^\alpha u(y)|$$ for each compact subset $K$ of $U$.

My question is: Are polynomials (or analytic functions) with values in $V$ dense in this space?

I could not find a definite answer so far...

Kofi
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    What does this mean? How can you view polynomials as taking values in V? – Owen Sizemore Aug 02 '13 at 09:26
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    For example by the tensor product. I don't think that the values here are so important. I would be interested in the result $V = \mathbb{R}$ as well. – Kofi Aug 02 '13 at 11:12
  • Have a look at http://math.stackexchange.com/questions/39957/are-polynomials-dense-in-ck-left-barb-right. Since every open ball around some $u$ only involves finitely many of those seminorms, the same argument implies that polynomials are dense in $C^\infty$. – Johannes Hahn Oct 14 '14 at 22:23

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