Lecture 19: Normal Families — Montel's Theorem

This file formalizes Theorem 1 from Lecture 19, which is Montel's theorem on normal families:

Let Ω ⊂ ℂ be a region, F a family of holomorphic functions on Ω such that for each compact E ⊂ Ω, F is uniformly bounded on E. Then F has a subsequence converging uniformly on each compact subset of Ω.

The proof follows the textbook argument:

1. Cauchy estimates give a Lipschitz bound for holomorphic functions on balls, which implies equicontinuity of the family on compact subsets.
2. A diagonal argument (Bolzano–Weierstrass on a countable dense set) extracts a subsequence converging pointwise on a dense subset of Ω.
3. The equicontinuity upgrades pointwise convergence on a dense set to uniform convergence on every compact subset of Ω (the "3ε argument").
4. The limit function is holomorphic by TendstoLocallyUniformlyOn.differentiableOn.

Step 1: Lipschitz bound from Schwarz estimates

theorem ball_subset_cthickening_of_mem {X : Type u_1} [PseudoMetricSpace X] {K : Set X} {z : X} (hz : z ∈ K) {δ : ℝ} : Metric.ball z δ ⊆ Metric.cthickening δ K

For z ∈ K and w ∈ ball(z, δ), we have w ∈ cthickening δ K.

theorem lipschitz_quarter_ball {R M : ℝ} (hR : 0 < R) {f : ℂ → ℂ} {c : ℂ} (hf : DifferentiableOn ℂ f (Metric.ball c R)) (hbnd : ∀ z ∈ Metric.ball c R, ‖f z‖ ≤ M) {z₁ z₂ : ℂ} (hz₁ : z₁ ∈ Metric.ball c (R / 4)) (hz₂ : z₂ ∈ Metric.ball c (R / 4)) : dist (f z₂) (f z₁) ≤ 4 * M / R * dist z₂ z₁

Holomorphic functions bounded on a ball satisfy a Lipschitz-type estimate on the quarter-ball. For z₁, z₂ ∈ ball(c, R/4), dist(f z₂, f z₁) ≤ (4M/R) · dist(z₂, z₁).

Step 2: Diagonal extraction

theorem diagonal_extraction_nat (x : ℕ → ℕ → ℂ) (hbdd : ∀ (k : ℕ), ∃ (M : ℝ), ∀ (n : ℕ), ‖x n k‖ ≤ M) : ∃ (φ : ℕ → ℕ), StrictMono φ ∧ ∀ (k : ℕ), ∃ (a : ℂ), Filter.Tendsto (fun (n : ℕ) => x (φ n) k) Filter.atTop (nhds a)

Diagonal extraction lemma. Given a doubly-indexed sequence x n k in ℂ that is bounded in the first index for each fixed second index, there exists a strictly increasing subsequence φ such that x (φ n) k converges for every k.

Step 3: Main theorem

theorem montel_normal_families {Ω : Set ℂ} (hΩ : IsOpen Ω) {f : ℕ → ℂ → ℂ} (hf : ∀ (n : ℕ), DifferentiableOn ℂ (f n) Ω) (hbdd : ∀ K ⊆ Ω, IsCompact K → ∃ (M : ℝ), ∀ (n : ℕ), ∀ z ∈ K, ‖f n z‖ ≤ M) : ∃ (φ : ℕ → ℕ) (g : ℂ → ℂ), StrictMono φ ∧ TendstoLocallyUniformlyOn (f ∘ φ) g Filter.atTop Ω ∧ DifferentiableOn ℂ g Ω

Montel's theorem (Theorem 1, Lecture 19).

Let Ω ⊆ ℂ be open, and let (fₙ) be a sequence of holomorphic functions on Ω that is locally uniformly bounded: for every compact K ⊆ Ω there exists M such that ‖fₙ(z)‖ ≤ M for all n and z ∈ K.

Then there exist a subsequence φ and a holomorphic function g on Ω such that f ∘ φ converges to g locally uniformly on Ω.