700 CHAPTER 24. THE BOCHNER INTEGRAL
for all n ̸= m and the norm refers to Lp ([a,b] ;W ). Let
a = t0 < t1 < · · ·< tk = b, ti− ti−1 = (b−a)/k.
Now define un (t)≡∑ki=1 uniX[ti−1,ti) (t) , uni ≡ 1
ti−ti−1
∫ titi−1
un (s)ds. The idea is to show thatun approximates un well and then to argue that a subsequence of the {un} is a Cauchysequence yielding a contradiction to 24.51.
Therefore,
un (t)−un (t) =k
∑i=1
un (t)X[ti−1,ti) (t)−k
∑i=1
uniX[ti−1,ti) (t)
=k
∑i=1
1ti− ti−1
∫ ti
ti−1
un (t)dsX[ti−1,ti) (t)−k
∑i=1
1ti− ti−1
∫ ti
ti−1
un (s)dsX[ti−1,ti) (t)
=k
∑i=1
1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))dsX[ti−1,ti) (t) .
It follows from Jensen’s inequality, Lemma 10.15.1, that
∥un (t)−un (t)∥pW =
k
∑i=1
∥∥∥∥ 1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))ds∥∥∥∥p
WX[ti−1,ti) (t)
And so ∫ T
0∥un (t)−un (t)∥p
W dt =k
∑i=1
∫ ti
ti−1
∥∥∥∥ 1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))ds∥∥∥∥p
Wdt
≤k
∑i=1
∫ ti
ti−1
ε
∥∥∥∥ 1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))ds∥∥∥∥p
Edt
+Cε
k
∑i=1
∫ ti
ti−1
∥∥∥∥ 1ti− ti−1
∫ ti
ti−1
(un (t)−un (s))ds∥∥∥∥p
Xdt (24.52)
Consider the second of these. It equals Cε ∑ki=1∫ ti
ti−1
∥∥∥ 1ti−ti−1
∫ titi−1
∫ ts u′n (τ)dτds
∥∥∥p
Xdt.This is
no larger than
≤ Cε
k
∑i=1
∫ ti
ti−1
(1
ti− ti−1
∫ ti
ti−1
∫ ti
ti−1
∥∥u′n (τ)∥∥
X dτds)p
dt
= Cε
k
∑i=1
∫ ti
ti−1
(∫ ti
ti−1
∥∥u′n (τ)∥∥
X dτ
)p
dt =Cε
k
∑i=1
((ti− ti−1)
1/p∫ ti
ti−1
∥∥u′n (τ)∥∥
X dτ
)p
Since b−ak = ti− ti−1,
= Cε
(k
∑i=1
(b−a
k
)1/p ∫ ti
ti−1
∥∥u′n (τ)∥∥
X dτ
)p
≤ Cε (b−a)k
(k
∑i=1
∫ ti
ti−1
∥∥u′n (τ)∥∥
X dτ
)p
=Cε (b−a)
k
(∥∥u′n∥∥
L1([a,b],X)
)p<
η p
10p