54.5.4 A Brief Review
First recall the definition of the metric on
For convenience it is listed here again.
Consider the unit sphere, S2
Define a map from the
complex plane to the surface of this sphere as follows. Extend a line from the
in the complex plane to the point
on the top of this sphere
denote the point of this sphere which the line intersects. Define
Then θ−1 is sometimes called sterographic projection. The mapping θ is clearly
continuous because it takes converging sequences, to converging sequences. Furthermore,
it is clear that θ−1 is also continuous. In terms of the extended complex plane,
converges to ∞
if and only if θzn
and a sequence,
converges to z ∈ ℂ
if and only if θ
In fact this makes it easy to define a metric on
Definition 54.5.9 Let z,w ∈
. Then let d
where this last
distance is the usual distance measured in ℝ3.
is a compact, hence complete metric space.
is a sequence in
. This means
is a sequence in
which is compact. Therefore, there exists a subsequence,
and a point,
z ∈ S2
that θznk → θz
which implies immediately that d
A compact metric
space must be complete.
Also recall the interesting fact that meromorphic functions are continuous with values
which is reviewed here for convenience. It came from the theory of classification of
Theorem 54.5.11 Let Ω be an open subset of ℂ and let f : Ω →
meromorphic. Then f is continuous with respect to the metric, d on
Proof: Let zn → z where z ∈ Ω. Then if z is a pole, it follows from Theorem 50.7.11
If z is not a pole, then f
Recall that θ
is continuous on ℂ
The fundamental result behind all the theory about to be presented is the Ascoli
Arzela theorem also listed here for convenience.
Definition 54.5.12 Let
be a complete metric space. Then it is said to be
locally compact if B
is compact for each r >
Thus if you have a locally compact metric space, then if
is a bounded sequence,
it must have a convergent subsequence.
Let K be a compact subset of ℝn and consider the continuous functions which have
values in a locally compact metric space,
denotes the metric on X
this space as C
Definition 54.5.13 For f,g ∈ C
, where K is a compact subset of ℝn and X is a
locally compact complete metric space define
The Ascoli Arzela theorem, Theorem 6.4.4 is a major result which tells which subsets
are sequentially compact.
Definition 54.5.14 Let A ⊆ C
for K a compact subset of ℝn. Then A is said to
be uniformly equicontinuous if for every ε >
0 there exists a δ >
0 such that whenever
x,y ∈ K with
< δ and f ∈ A,
The set, A is said to be uniformly bounded if for some M < ∞, and a ∈ X,
for all f ∈ A and x ∈ K.
The Ascoli Arzela theorem follows.
Theorem 54.5.15 Suppose K is a nonempty compact subset of ℝn and A ⊆ C
is uniformly bounded and uniformly equicontinuous where X is a locally compact complete
metric space. Then if
⊆ A, there exists a function, f ∈ C
and a subsequence,
fkl such that
In the cases of interest here, X =
with the metric defined above.