# Vitali's Theorem on convergence of holomorphic functions

Can anybody point me to a statement and proof of Vitali's (convergence) Theorem, which states (I think) that nets of holomorphic functions converge under certain conditions?

I understand that there are different formulations of this theorem. I'm looking for a reference, which treats the theorem comprehensively.

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Here is what he calls Vitali-Porter theorem: Let $\{f_n\}$ be a locally bounded sequence of analytic functions in a domain $\Omega$ such that the limit $\lim_{n\to \infty}f_n(z)$ exists for all $z$ belonging to a subset $E \subset \Omega$ which has an accumulation point in $\Omega$. Then $f_n$ converges uniformly on compact subsets of $\Omega$ to an analytic function.
You can assume wlog that $E$ is countable. If $\{f_\alpha\}_{\alpha \in A}$ is a net of functions that does not converge uniformly on a set $K$, but does converge pointwise on $E$, it is easy to extract a subsequence that does not converge uniformly on $K$ but does converge pointwise on $E$. So the sequence version does imply the net version. – Robert Israel Dec 6 '11 at 20:41