55.5. THE PICARD THEOREMS 1761

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 55.5.12 Let (X ,d) be a complete metric space. Then it is said to be locallycompact if B(x,r) is compact for each r > 0.

Thus if you have a locally compact metric space, then if {an} is a bounded sequence, itmust have a convergent subsequence.

Let K be a compact subset of Rn and consider the continuous functions which havevalues in a locally compact metric space, (X ,d) where d denotes the metric on X . Denotethis space as C (K,X) .

Definition 55.5.13 For f ,g ∈ C (K,X) , where K is a compact subset of Rn and X is alocally compact complete metric space define

ρK ( f ,g)≡ sup{d ( f (x) ,g(x)) : x ∈ K} .

The Ascoli Arzela theorem, Theorem 7.8.4 is a major result which tells which subsetsof C (K,X) are sequentially compact.

Definition 55.5.14 Let A ⊆ C (K,X) for K a compact subset of Rn. Then A is said to beuniformly equicontinuous if for every ε > 0 there exists a δ > 0 such that whenever x,y∈Kwith |x−y|< δ and f ∈ A,

d ( f (x) , f (y))< ε.

The set, A is said to be uniformly bounded if for some M < ∞, and a ∈ X ,

f (x) ∈ B(a,M)

for all f ∈ A and x ∈ K.

The Ascoli Arzela theorem follows.

Theorem 55.5.15 Suppose K is a nonempty compact subset of Rn and A ⊆ C (K,X) , isuniformly bounded and uniformly equicontinuous where X is a locally compact completemetric space. Then if { fk} ⊆ A, there exists a function, f ∈ C (K,X) and a subsequence,fkl such that

liml→∞

ρK(

fkl , f)= 0.

In the cases of interest here, X = Ĉ with the metric defined above.

55.5.5 Montel’s TheoremThe following lemma is another version of Montel’s theorem. It is this which will makepossible a proof of the big Picard theorem.