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The complex Stone-Weierstrauss theorem can be applied to compact spaces. However, say I have an open set $D\subset \mathbb{C}$, and I want to approximate a holomorphic function $f$ on $D$ via polynomials. For every compact subset $K$ of $D$, the Stone-Weierstrauss theorem guarantees that a sequence of polynomials will converge uniformly on $K$ to $f$, since $f$ is continuous.

Then, is it guaranteed that there exists a common sequence of polynomials on $D$ that will converge to $f$ uniformly on compact subsets of $D$?

Thanks!

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Write $D$ as a increasing sequence of compacts $K_n$ (how)? For each $n$ take $P_n$ with $|f-P_n|\le 1/n$ in $K_n$.

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  • $\begingroup$ I didn't think of using inner regularity! That's a cute proof. Thanks very much :) $\endgroup$ – Merry Feb 21 '16 at 20:37

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