2386 CHAPTER 69. GELFAND TRIPLES

Proof of claim: Let ε > 0 be given. Let t ∈ [a,b] . Pick tm ∈ D∩ [a,b] such that in69.5.27 Cε R |t− tm|< ε/3. 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 69.5.27

||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. There exists N such that if m,n>N, then ||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 (69.5.28)

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, 69.5.27 shows themiddle term in 69.5.28 is also smaller than ε. Therefore, if |t− s| is small enough,

||u(t)−u(s)||W < 3ε.

Thus u is continuous. Finally, let N be as in the above claim. Then letting m,n > N, itfollows that for all t ∈ [a,b] ,

||um (t)−un (t)||W < ε.