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Montel's theorem: every locally bounded holomorphic family is normal

Statement

Assume the Axiom of Choice, and therefore in particular Countable Choice and Dependent Choice for the successive subsequence selections. Let $Ω$ be a plane domain and let $F \subseteq H (Ω)$ be locally bounded. Then $F$ is a normal family.

Facts & Assumptions

Given: Choice, a plane domain $Ω$ , and a locally bounded family $F \subseteq H (Ω)$ .

[L1]

On a compact metric domain, compactness of the uniform closure is equivalent to equicontinuity and pointwise relative compactness (Ascoli–Arzelà in the uniform topology for nonempty compact metric domains).

[L2]

Holomorphic functions are closed under locally uniform limits (Holomorphic functions form a closed subspace for locally uniform convergence).

[L3]

The canonical exhaustion $(K_{n})$ is compact, nested, and has interiors covering $Ω$ (Every plane domain has the canonical nested compact exhaustion by distance and radius cutoffs).

[L4]

Locally bounded holomorphic families are locally equicontinuous (Locally bounded holomorphic families are locally equicontinuous).

[L5]

In a metric space, compactness is equivalent to sequential compactness under Countable Choice and Dependent Choice (For a metric space, compact, countably compact, limit point compact, sequentially compact, and complete together with totally bounded are all equivalent, given countable choice and dependent choice).

Proof

technique · direct

1.1

On each compact stage $K_{n}$ of [L3], local boundedness plus [L4] makes the restricted family equicontinuous and pointwise relatively compact; hence [L1] makes its uniform closure compact.

L1L3L4given

1.2

Using [L5], choose successively a subsequence converging uniformly on $K_{1}$ , then a further subsequence converging uniformly on $K_{2}$ , and so on, and take the diagonal subsequence. This is the explicit Choice step named in the Statement.

L5givenchoose

1.3

For each fixed $m$ , the diagonal subsequence eventually lies in the $m$ th chosen subsequence, so it converges uniformly on $K_{m}$ . Because the interiors of the $K_{m}$ cover $Ω$ by [L3], this gives local uniform convergence on all of $Ω$ .

L3given

2.1

Fact [L2] makes the local uniform limit holomorphic, so every sequence in $F$ has a locally uniformly convergent subsequence in $H (Ω)$ . That is exactly normality.

L2given∎

Depends on

Used by

Dependency tree · two levels

34 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)