TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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Normal holomorphic families are locally bounded

Statement

Assume the Axiom of Countable Choice. Every normal family of holomorphic functions on a plane domain is locally bounded.

Facts & Assumptions

Given: Countable Choice and a normal family $F \subseteq H (Ω)$ .

[L1]

Normality means that every sequence in $F$ has a subsequence converging locally uniformly to a holomorphic limit (Normal families of holomorphic functions on a plane domain).

Proof

technique · direct

1.1

If $F$ were not locally bounded at some point, then on some closed disc there would be a sequence $(f_{n})$ in $F$ with $\sup ∣ f_{n} ∣ > n$ .

givenchoose

2.1

By [L1], a subsequence would converge uniformly on that disc to a holomorphic limit, and uniform convergence on a compact disc forces that subsequence to be uniformly bounded there. This contradicts step 1.1, so the family is locally bounded.

L1givenalgebra∎

Depends on

Normal families of holomorphic functions on a plane domain

Used by

The family nz is not normal on any domain containing zero Counterexample
The family zⁿ is normal on the unit disc and not normal on the complex plane Example

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)