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

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