Normal family - Knowlpedia

Definition

Let $D\subseteq\mathbb C$ be a domain. A family $\mathcal F$ of holomorphic functions $D\to\mathbb C$ is a normal family if every sequence in $\mathcal F$ has a subsequence that converges uniformly on each compact subset of $D$ , either to a holomorphic function $D\to\mathbb C$ or locally uniformly to $\infty$ .

Spherical formulation

View each function as a map into the Riemann sphere with its spherical metric. Then the two alternatives in the core can be expressed as locally uniform spherical convergence. For families of meromorphic functions, this spherical formulation is the standard definition and permits meromorphic limits.

Compactness criterion

The Montel theorem says that every locally uniformly bounded family of holomorphic functions is normal. It is the central compactness criterion used in many existence proofs.

Role in conformal mapping

Normal-family compactness is used in a standard proof of the Riemann mapping theorem: one takes a sequence of normalized injective maps whose derivative at a base point approaches the extremal value, extracts a limit, and proves that extremality forces surjectivity.

References

John B. Conway, Functions of One Complex Variable I, 2nd ed., Springer, 1978. Publisher record. Relevant: Chapter VII.