696 CHAPTER 24. THE BOCHNER INTEGRAL

≤ ε

6R(∥u(t)∥E +∥u(s)∥E)+Cε ∥u(t)−u(s)∥X ≤

ε

3+Cε R |t− s|1/q . (24.47)

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 in 24.47Cε 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 24.47

∥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 itfollows that ∥un (s)−um (s)∥ < ε. Since [a,b] is compact, there are finitely many of theseballs, {B(ti,δ )}p

i=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 it follows

that ||un−um||∞,W < ε.Now let u(t) = limk→∞ uk (t) .

∥u(t)−u(s)∥W ≤ ∥u(t)−un (t)∥W +∥un (t)−un (s)∥W +∥un (s)−u(s)∥W (24.48)

Let N be in the above claim and fix n > N. Then

∥u(t)−un (t)∥W = limm→∞∥um (t)−un (t)∥W ≤ ε

and similarly, ∥un (s)−u(s)∥W ≤ ε. Then if |t− s| is small enough, 24.47 shows the middleterm in 24.48 is also smaller than ε. Therefore, if |t− s| is small enough,

∥u(t)−u(s)∥W < 3ε.