224 CHAPTER 9. BASIC FUNCTION SPACES
the process must stop since otherwise, you would have an infinite sequence of points withnot limit point because they are all ε apart. This contradicts the compactness of K. ■
Recall Lemma 9.1.2. If you consider C (K,Rn) it is automatically equal to BC (K,Rn)because of the extreme value theorem applied to x→ |f(x)| for x ∈ K. Therefore, the spaceC (K,Rn) is complete with respect to the norm defined there.
Definition 9.2.2 Let A be a set of functions in C (K,Rn) . It is called equicontin-uous if for every ε > 0 there exists δ > 0 such that if |x−y| < δ , then |f(x)− f(y)| < ε
for all f ∈ A . In words, the functions in A are uniformly continuous for all f at once. Aset A ⊆C (K,Rn) is uniformly bounded if there is a large enough positive number M suchthat max{|f(x)| : x ∈ K, f ∈A }< M.
The significant property of an equicontinuous set of functions is the following.
Lemma 9.2.3 If {gk}∞
k=1 is equicontinuous and converges pointwise to g on a compactset K, then the sequence converges uniformly on K.
Proof of claim: Let ε > 0 be given and let δ go with ε/4 in the definition of equicon-tinuous. By compactness and Proposition 9.2.1, there are finitely many points of K, de-noted as {x1, · · · ,xs} such that K ⊆∪s
i=1B(xi,δ ). There exists Ni such that if k, l ≥Ni, then|gl (xi)−gk (xi)|< ε
4 . Thus if N ≥max{Ni, i = 1, · · · ,s} , then for all xi, |gl (xi)−gk (xi)|<ε
4 if k ≥ N. Then for k, l ≥ N, and x arbitrary, let x ∈ B(xi,δ ) . Then
|gl (x)−gk (x)| ≤ |gl (x)−gl (xi)|+ |gl (xi)−gk (xi)|+ |gk (xi)−gk (x)|
<ε
4+
ε
4+
ε
4
Thus for k, l ≥N, ∥gl−gk∥∞< 3ε
4 < ε. This shows {gk} is a Cauchy sequence in C (K,Rn)which is complete. Thus this sequence converges uniformly to some g ∈C (K,Rn). ■
The following is the Arzela Ascoli theorem . Actually, the converse is also true but Iwill only give the direction of most use in applications.
Theorem 9.2.4 Let A ⊆C (K,Rn) be both equicontinuous and uniformly bounded.Then every sequence in A has a convergent subsequence converging to some g∈C (K,Rn),the convergence taking place with respect to ∥·∥
∞, the uniform norm.
Proof: Let{
f j}∞
j=1 be a sequence of functions in A . Let D be a countable densesubset of K. Say D≡ {dk}∞
k=1 . Then{
f j (d1)}∞
j=1 is a bounded set of points in Rn. By theHeine Borel theorem, there is a subsequence, denoted by
{f( j,1) (d1)
}∞
j=1 which converges.
Now apply what was just done with{
f j}
to{
f( j,1)}
and feature d2 instead of d1. Thus{f( j,2)
}∞
j=1 is a subsequence of{
f( j,1)}
which converges at d2. This new subsequence stillconverges at d1 thanks to Theorem 2.2.10. Continue this way. Thus we get the following
f(1,1) f(2,1) f(3,1) · · · converges at d1f(1,2) f(2,2) f(3,2) · · · converges at d1,d2f(1,3) f(2,3) f(3,3) · · · converges at d1,d2,d3
......
......
f(1,l) f(2,l) f(3,l) · · · converges at d j, j ≤ l...
......
...