Normal family
Definition. A family \(\mathcal F\) of holomorphic functions on \(U\) is normal if every sequence in \(\mathcal F\) has a subsequence converging locally uniformly, possibly to \(\infty\) in the spherical metric when meromorphic functions are allowed.
Montel theorem
Theorem. A locally uniformly bounded family of holomorphic functions is normal. Cauchy's estimates give local equicontinuity, and Arzelà-Ascoli gives subsequential compactness on an exhaustion by compact subsets.
Derivative bounds
Lemma. If \(|f|\le M\) on \(D(a,R)\), then \(|f'|\le C M/R\) on smaller discs. These uniform derivative bounds are the equicontinuity input in Montel's theorem.
Vitali theorem
Theorem. If a locally bounded sequence of holomorphic functions converges pointwise on a set with an accumulation point, then it converges locally uniformly to a holomorphic function. Normality supplies subsequential limits, and identity forces all limits to agree.
Extremal maps in Riemann mapping
Example. The extremal proof of the Riemann mapping theorem uses Montel's theorem to extract a holomorphic limit from a maximizing sequence of normalized maps.
Compactness philosophy
Remark. Normal families are compactness for holomorphic functions. Cauchy estimates turn boundedness into enough regularity for subsequential limits.
Normality and compact-open convergence
Definition. A family \(\mathcal F\subset\mathcal O(\Omega)\) is normal if every sequence in \(\mathcal F\) has a subsequence converging locally uniformly either to a holomorphic function or, in the spherical formulation for meromorphic functions, to \(\infty\).
Montel's theorem
Theorem (Montel). A locally uniformly bounded family of holomorphic functions on a domain is normal.
Proof. On every compact \(K\Subset\Omega\), choose a slightly larger compact neighborhood on which \(|f|\le M\) uniformly. Cauchy's estimate then bounds \(|f'|\) uniformly on \(K\), making the family equicontinuous. Arzelà--Ascoli gives a uniformly convergent subsequence on each compact. Exhaust \(\Omega\) by nested compacts and take a diagonal subsequence. The locally uniform limit is holomorphic by Weierstrass's theorem.
Vitali convergence
Theorem (Vitali). Let \(f_n\) be a locally bounded sequence of holomorphic functions. If \(f_n(z)\) converges on a set with an accumulation point in \(\Omega\), then \(f_n\) converges locally uniformly on \(\Omega\) to a holomorphic function.
Proof. Montel makes every subsequence have a locally uniformly convergent further subsequence. Any two such limits agree on the pointwise convergence set and hence everywhere by the identity theorem. Relative compactness plus uniqueness of every cluster point implies the full sequence converges.
Hurwitz and preservation of injectivity
Proposition. A locally uniform limit of injective holomorphic functions is either injective or constant.
Proof. If the nonconstant limit \(f\) had \(f(z_1)=f(z_2)\), choose disjoint small discs around the two points. For nearby \(w\), Rouché or Hurwitz shows \(f_n-w\) has a zero in each disc for large \(n\), contradicting injectivity.
Marty and Zalcman viewpoints
For meromorphic families, local boundedness is replaced by control of the spherical derivative \[ f^\#(z)=\frac{|f'(z)|}{1+|f(z)|^2}. \] Marty's criterion says local boundedness of \(f^\#\) is equivalent to normality. Zalcman's lemma says failure of normality produces rescaled maps converging to a nonconstant entire or meromorphic function. This converts local compactness failure into a global function, making Picard-type arguments possible.