Math 411: Honours Complex Variables - University of Alberta

Define g: D → RM as follows: for x ∈ D, let k be the smallest natural number
such that x ∈ Kk, set g(x) := gk(x). Then fn,n|Kk→ g|Kk uniformly on Kk.
Let K ⊂ D be compact. By the choices of K1,K2,..., we have K ⊂ D =
� ∞
k=1 intKk, so that {intKk : k ∈ N} is an open cover for K. Since K is compact,
and since intKk ⊂ intKk+1 for k ∈ N, there exists k0 ∈ N such that K ⊂ intKk0 ⊂
Kk0. Since fn,n|Kk 0 → g|Kk 0 uniformly on Kk0, it follows that fn,n|K→ g|K uniformly
on K.
Theorem 16.2 (Montel’s Theorem). Let D ⊂ C be open, and let F be a family of
holomorphic functions on D that is uniformly bounded on compact subsets of D. Then
every sequence in F has a subsequence that converges compactly to a holomorphic
function on D.
Proof. In view of Proposition 16.1, we only need to show that F is equicontinuous
on compact subsets of D.
Let z0 ∈ D, and let r > 0 be such that B2r[z0] ⊂ D. There exists C > 0 such that
|f(ζ)|≤ C for all f ∈ F and all ζ ∈ ∂B2r(z0).
Let f ∈ F, and let z,w ∈ Br(z0). Then we have:
|f(z)−f(w)| = 1
��
�
�
2π �
= 1
��
�
�
2π �
= |z −w|
2π
∂B2r(z0)
∂B2r(z0)
��
�
�
�
�
�
dζ�
�
�
�
dζ�
�
f(ζ)
(ζ −z)(ζ −w) dζ
�
�
�
�
� �
f(ζ) f(ζ)
−
ζ −z ζ −w
f(ζ)(ζ −w +z −ζ)
(ζ −z)(ζ −w)
∂B2r(z0)
|z −w|
≤
2π 4πrC
r2 = 2C
|z −w|.
r
For ǫ > 0, choose δ := rǫ
2C , so that |f(z) − f(w)|< ǫ for all z,w ∈ Br(z0) with
|z −w|< δ.
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