698 CHAPTER 21. THE BOCHNER INTEGRAL
Since ε is arbitrary, it follows u ∈C ([a,b] ;W ).Let D = Q∩ [a,b] so D is a countable dense subset of [a,b]. Let D = {tn}∞
n=1. Bycompactness of the embedding of E into W, there exists a subsequence u(n,1) such thatas n→ ∞, u(n,1) (t1) converges to a point in W. Now take a subsequence of this, called(n,2) such that as n→ ∞,u(n,2) (t2) converges to a point in W. It follows that u(n,2) (t1) alsoconverges to a point of W. Continue this way. Now consider the diagonal sequence, uk ≡u(k,k) This sequence is a subsequence of u(n,l) whenever k > l. Therefore, uk (t j) convergesfor all t j ∈ D.
Claim: Let {uk} be as just defined, converging at every point of D ≡ [a,b]∩Q. Then{uk} converges at every point of [a,b].
Proof of claim: Let ε > 0 be given. Let t ∈ [a,b] . Pick tm ∈ D∩ [a,b] such that in21.10.47 Cε R |t− tm|< ε/3. Then there exists N such that if l,n > N, then
||ul (tm)−un (tm)||X < ε/3.
It follows that for l,n > N,
||ul (t)−un (t)||W ≤ ||ul (t)−ul (tm)||W + ||ul (tm)−un (tm)||W+ ||un (tm)−un (t)||W
≤ 2ε
3+
ε
3+
2ε
3< 2ε
Since ε was arbitrary, this shows {uk (t)}∞
k=1 is a Cauchy sequence. Since W is complete,this shows this sequence converges.
Now for t ∈ [a,b] , it was just shown that if ε > 0 there exists Nt such that if n,m > Nt ,then
||un (t)−um (t)||W <ε
3.
Now let s ̸= t. Then
||un (s)−um (s)||W ≤ ||un (s)−un (t)||W + ||un (t)−um (t)||W + ||um (t)−um (s)||W
From 21.10.47
||un (s)−um (s)||W ≤ 2(
ε
3+Cε R |t− s|1/q
)+ ||un (t)−um (t)||W
and so it follows that if δ is sufficiently small and s ∈ B(t,δ ) , then when n,m > Nt
||un (s)−um (s)||< ε.
Since [a,b] is compact, there are finitely many of these balls, {B(ti,δ )}pi=1 , such that for
s ∈ B(ti,δ ) and n,m > Nti , the above inequality holds. Let N > max{
Nt1 , · · · ,Ntp
}. Then
if m,n > N and s ∈ [a,b] is arbitrary, it follows the above inequality must hold. Therefore,this has shown the following claim.
Claim: Let ε > 0 be given. Then there exists N such that if m,n > N, then
||un−um||∞,W < ε.