DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-28

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Normal families of holomorphic functions on a plane domain

Definition

Let $Ω \subseteq C$ be a plane domain and let $F$ be a family of holomorphic functions on $Ω$ . The family $F$ is normal when every sequence $(f_{n})$ in $F$ has a subsequence $(f_{n_{k}})$ and a holomorphic function $f$ on $Ω$ such that $f_{n_{k}} \to f$ locally uniformly on $Ω$ .

By Locally uniform convergence on a plane domain is the already-published compact-convergence notion, the same condition can be read as uniform convergence on every compact subset of $Ω$ .

Depends on

Locally uniform convergence on a plane domain is the already-published compact-convergence notion

Used by

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Matthias Weber, Complex Analysis, Ch. 5 §§5.1-5.2 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 2 §5.2 and Ch. 8 §3.2 (standard reference, not scraped)
Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, Ch. 2 (standard reference, not scraped)