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I was reviewing my complex analysis, and found this problem in a problem set. It says "prove that every holomorphic function on the disc $D=\{|z|<1\}$ is a uniform limit of polynomials".

I'm confused about it, it seems to me that the statement as it is is not true. It should be true that the convergence is uniform in every compact set contained in the disc though. However, for instance, I think it is not possible to approximate the function $\frac{1}{z-1}$ with polynomials that converge uniformly. Am I right?

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2 Answers 2

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You are right, and your example shows that not every holomorphic function on the open unit disk is the uniform limit of polynomials.

However, every function that is continuous on the closed unit disk and holomorphic on the open unit disk is the uniform limit of polynomials, and every holomorphic function on the open unit disk is the locally uniform limit of polynomials. (That of course generalises to arbitrary disks.)

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  • $\begingroup$ Where can I find a proof of this fact? $\endgroup$ Aug 17, 2021 at 13:37
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Of course you are right: it is impossible to approximate an unbounded function uniformly with bounded functions.

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