Montel's theorem

Montel's theorem states that every locally bounded family of holomorphic functions on a domain is normal: every sequence has a subsequence converging uniformly on compact subsets, in the spherical metric if the limit may be meromorphic.

Normal family: A family of meromorphic functions in which every sequence has a subsequence converging locally uniformly in the spherical metric. Montel's theorem turns local boundedness of holomorphic families into normality.
Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. Holomorphicity supplies the analytic estimates that make local boundedness unusually restrictive.
Vitali convergence theorem: A theorem upgrading pointwise convergence on a suitable set to locally uniform convergence for a normal holomorphic family. Montel's theorem supplies the normality needed for Vitali's convergence criterion.
Paul Montel: A French mathematician who developed the theory of normal families of holomorphic functions. His work established the framework named by Montel's theorem.
Arzelà–Ascoli theorem: A compactness theorem giving uniformly convergent subsequences for equicontinuous, uniformly bounded families of functions. Cauchy estimates provide equicontinuity, allowing compactness arguments on compact subsets.
Domain (complex analysis): A connected open subset of the complex plane. The theorem applies on domains and expresses convergence on their compact subsets.
Riemann mapping theorem: A theorem stating that every simply connected proper plane domain is conformally equivalent to the unit disk. A normal-family compactness argument is central to a classical proof of the mapping theorem.
Normal families in complex analysis: The study of families of holomorphic or meromorphic functions whose sequences admit locally convergent subsequences. Montel's theorem is a foundational compactness result in this theory.
Cauchy's integral formula: A formula expressing a holomorphic function's interior values through its values along a surrounding contour. Its derivative estimates convert local bounds into uniform control of variation.
Locally bounded function family: A family of functions bounded uniformly on some neighborhood of each point in its domain. This is the precise hypothesis that triggers Montel's compactness conclusion.

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Émile PicardRelated: Picard’s value-omission ideas became a key ingredient in normal-family theory.