Alphabeta MathHow statement and proof provenance work
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FALSE: Arzelà-Ascoli alone proves Montel's theorem
Statement
Arzelà-Ascoli alone proves Montel's theorem for holomorphic families.
Facts & Assumptions
Given: A locally bounded family of holomorphic functions on a plane domain.
On a compact metric domain, Arzelà–Ascoli requires equicontinuity as well as pointwise relative compactness (Ascoli–Arzelà in the uniform topology for nonempty compact metric domains).
Local boundedness of a holomorphic family implies local equicontinuity by Cauchy estimates (Locally bounded holomorphic families are locally equicontinuous).
Refutation
Local boundedness gives pointwise relative compactness on each compact stage, but it is not itself the equicontinuity hypothesis required by [L1]. Thus Arzelà–Ascoli cannot yet be applied.
Fact [L2] supplies the missing complex-analytic step by converting local boundedness into local equicontinuity. Montel's proof uses that step before applying Arzelà–Ascoli, so Arzelà–Ascoli alone does not prove the theorem.
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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)