Vitali's theorem
/ vee-TAH-lee /
Suppose you have a sequence of holomorphic functions that is bounded (so Montel guarantees convergent SUBSEQUENCES), but you would really like to know the WHOLE sequence converges, not just some subsequence. In general a bounded sequence can have many subsequences running to different limits. Vitali's theorem gives a wonderfully cheap way to rule that out: if a locally bounded sequence of holomorphic functions converges on just a small 'witness' set — a set with a limit point inside the domain — then it converges on the entire domain.
Precisely: let f_n be holomorphic on a connected region D and locally bounded there. Suppose f_n(z) converges (as a sequence of numbers) for every z in some subset E of D, where E has at least one accumulation point lying in D. Then f_n converges locally uniformly on all of D, and its limit is holomorphic. The proof is the marriage of Montel and the identity theorem: by Montel the family is normal, so any two convergent subsequences have holomorphic limits; those two limits agree on E, which has an accumulation point, so by the identity theorem they agree everywhere on D. Since every subsequence has the same limit, the full sequence converges. A handy special case is that convergence on a tiny disk, or even on a single convergent arc, already forces convergence everywhere.
Vitali's theorem is the tool that promotes 'subsequential' compactness to genuine convergence, and it is the standard way to prove that an iterative or perturbative scheme of holomorphic functions actually has a limit. It is also a rigidity statement in disguise: holomorphic functions are so tightly determined that agreement of limits on a thin set propagates globally. Honest caveat: the witness set E must have a limit point IN the domain — convergence merely on a sequence of points that escapes to the boundary, or on an isolated set with no accumulation point, is not enough. And local boundedness is essential; without it Montel fails and the conclusion can collapse.
Let f_n be holomorphic and bounded by 1 on the unit disk, and suppose f_n(1/k) converges for every k = 1, 2, 3, ... . The points 1/k accumulate at 0, which lies inside the disk, so Vitali's theorem says f_n converges locally uniformly on the entire disk — even though we only checked convergence along one shrinking sequence of real points.
Convergence on a set with an interior limit point (here the 1/k) propagates to the whole connected domain.
The witness set must accumulate at an interior point of the domain; isolated points or a set clustering only on the boundary will not do. The theorem rests on Montel (local boundedness) plus the identity theorem, so connectedness of the domain is also required.