Montel's Theorem states that a family of holomorphic functions on an open subset of the complex plane that is uniformly bounded on every compact subset is a normal family, meaning every sequence drawn from it has a subsequence converging locally uniformly on the domain. Named for Paul Montel, it is a foundational compactness result of complex analysis used throughout the theory of conformal mapping and complex dynamics.
Facts
StatementA family of holomorphic functions on an open subset of the complex numbers is normal if and only if it is locally uniformly bounded. 1 Classification
Statement FormCharacterization Theorem 1 Sources
1. Montel's theorem (Wikipedia)
Locally uniformly bounded families are normalQuote, Locally uniformly bounded families are normal
A family of holomorphic functions defined on an open subset of the complex numbers is normal if and only if it is locally uniformly bounded.
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