Math 411: Honours Complex Variables - University of Alberta

Chapter 16
Montel’s Theorem
Definition. LetS ⊂ R N . AfamilyF offunctionsonS intoR M iscalledequicontinuous
if, for each ǫ > 0, there exists δ > 0 such that |f(x)−f(y)|< ǫ for all f ∈ F and for
all x,y ∈ S such that |x−y|< δ.
Lemma 16.1. Let S ⊂ R N . Then S contains a countable dense subset.
Proof. Let {x1,x2,x3,...} be a dense, countable subset ofR N , e.g., Q N . For n,m ∈ N
with S ∩B1 (xn) �= ∅, choose yn,m ∈ S ∩B1
m
m
�
yn,m : n,m ∈ N,S ∩B1
m
(xn). Then
�
(xn) �= ∅ ⊂ S
is countable.
Let ǫ > 0 and x ∈ S. Choose m ∈ N so large that 1 ǫ < m 2 . Since {x1,x2,x3,...} is
dense in RN , there exists n ∈ N such that |xn−x|< 1
and thus x ∈ S∩B 1 (xn) �= ∅.
m m
It follows that
|yn,m−x|≤ |yn,m−xn|+|xn −x|< 2
< ǫ
m
Theorem 16.1 (Arzelà–Ascoli Theorem). Let K ⊂ R N be compact, and let F be
an equicontinuous and uniformly bounded family of functions from K to R M . Then
every sequence in F has a subsequence that converges uniformly on K.
Proof. Let (fn) ∞ n=1 be a sequence in F, and let {x1,x2,x3,...} be a countable dense
subset of K.
Since(fn(x1)) ∞ n=1 isaboundedsequenceinR M ,thereexistsasubsequence(fn,1) ∞ n=1
of (fn) ∞ n=1 such that (fn,1(x1)) ∞ n=1 converges.
Since (fn,1(x2)) ∞ n=1 is a bounded sequence in RM , there exists a subsequence
(fn,2) ∞ n=1 of (fn,1) ∞ n=1 such that (fn,2(x2)) ∞ n=1 converges.
Continuing inductively in this fashion, we obtain, for each k ∈ N, a subsequence
(fn,k) ∞ n=1 of (fn) ∞ n=1 such that, for each k ∈ N,
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