# Do you know the name of this theorem

and perhaps a reference.

Given a complete normed space $S$ (metric space may be sufficient, I'm not sure) with a compact subset $C$ and a function $f$ that is analytic on $C$ then if $f(x)=0$ for infinitely many $x\in C$ then $f$ is identically 0 on $C$.

I'm sure I learned this in an undergraduate analysis class but for the moment it has escaped me. I have found it in reference to p-adic numbers so I would like to know the theorem for as general a case as possible but it you only know of the theorem for just the reals please tell me.

Also if my hypothesis is lacking please tell me that too.

Thanks
Steven

## Answers and Replies

I don't know if this helps, but Liouville's theorem tells us that if a function is bounded and entire on the complex plane then it is identically constant.

mathwonk
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its false as stated. you seem to want some form of analytic continuation, which involves connectivity

Hmm, connectivity does seem to be an issue. A friend pointed out to me that this is based on the analytic identity theory which basically says if two functions are analytic on a domain and they agree on a set with an accumulation point then they are equal on the domain. Connectivity seems to be a hairy point though. I don't think Zp is path connected in Qp. It does have domain like properties though so their should be a way to apply things.

Thanks guys,
Steven

mathwonk
Science Advisor
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2020 Award
you may not need path connected, just connected.

by thew way analytic means given by a power series:

anz^n + .... = z^n (an + ....).

it follows that if an is not zero, then since there is an open nbhd where the second factor is not zero, that the power series has an isolated zero at z=0.

thism is true near any zero, so on any open set, the zeroes of an analytic function are isolated, unless the fucntion is identically zero in a nbhd of that point, hence on the connected component containing that point.

hence if f,g are two different analytic functions in some open set, then the zeroes of f-g are isolated. hence if f = g in some set that has an accumulation point, then the difference is identically zero in the connected component of their common open domain, containing thata ccumualtion point.

something like that anyway. this should be in any complex analysis book.

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