Montel's theorem
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Montel's theorem is the practical key that unlocks normal families. Checking the definition of a normal family directly — 'every sequence has a locally uniformly convergent subsequence' — sounds hard, because you would have to produce subsequences by hand. Montel's theorem hands you a simple, checkable condition that does all that work for you: if a family of holomorphic functions is uniformly bounded on each compact piece of the domain, then it is automatically normal. Boundedness alone buys you compactness.
Precisely: a family F of holomorphic functions on an open region D that is locally bounded — meaning for every compact subset K there is a single constant M with |f(z)| <= M for all z in K and all f in F — is a normal family. The proof is a marriage of Cauchy and Arzela-Ascoli. The Cauchy integral formula turns the bound on |f| into a bound on |f'| on a slightly smaller set: if |f| <= M on a disk of radius R, then |f'| <= M/R nearby. A uniform bound on derivatives means the functions cannot wiggle too fast, which is exactly equicontinuity. With local boundedness plus equicontinuity in hand, the Arzela-Ascoli theorem delivers the locally uniformly convergent subsequence, and the Weierstrass theorem guarantees its limit is holomorphic. So the chain is: bounded values -> bounded derivatives -> equicontinuity -> compactness.
This is the version used over and over in conformal mapping: the candidate maps in the Riemann mapping theorem all land in the unit disk, hence are bounded by 1, hence form a normal family for free. There is also a far deeper relative — the 'fundamental normality test' — which gets normality from merely OMITTING two values rather than from a numerical bound. Honest caveat: Montel's theorem is a holomorphic phenomenon. The same statement is FALSE for arbitrary smooth or continuous functions; it is the rigidity of holomorphy (through the Cauchy estimates) that converts a bound on size into control of all derivatives, and that is what makes compactness appear.
Consider the family F = { f holomorphic on the unit disk with |f(z)| <= 5 for all z }. On the smaller disk |z| <= 1/2, the Cauchy estimate gives |f'(z)| <= 5 / (1/2) = 10 for every f in F. That uniform derivative bound forces equicontinuity, so by Montel's theorem any sequence from F has a locally uniformly convergent subsequence.
A uniform bound on values yields a uniform bound on derivatives (Cauchy), hence equicontinuity, hence normality.
Local boundedness is sufficient but not necessary for normality — a family can be normal yet unbounded (e.g. one allowing the limit infinity). The theorem fails outside holomorphy: bounded continuous functions need not be equicontinuous.