698 CHAPTER 24. THE BOCHNER INTEGRAL
Proof: It suffices to show S has an η net in Lp ([a,b] ;W ) for each η > 0.If not, there exists η > 0 and a sequence {un} ⊆ S, such that
∥un−um∥ ≥ η (24.49)
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 the above ∥un−um∥ ≥ η .
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)
≤k
∑i=1
1ti− ti−1
∫ ti
ti−1
∥un (t)−un (s)∥pW dsX[ti−1,ti) (t)
and so ∫ b
a∥un (t)−un (s)∥p
W ds
≤∫ b
a
k
∑i=1
1ti− ti−1
∫ ti
ti−1
∥un (t)−un (s)∥pW dsX[ti−1,ti) (t)dt
=k
∑i=1
1ti− ti−1
∫ ti
ti−1
∫ ti
ti−1
∥un (t)−un (s)∥pW dsdt. (24.50)
From Lemma 24.11.1 if ε > 0, there exists Cε such that
∥un (t)−un (s)∥pW ≤ ε ∥un (t)−un (s)∥p
E +Cε ∥un (t)−un (s)∥pX
≤ 2p−1ε (∥un (t)∥p +∥un (s)∥p)+Cε |t− s|p/q
This is substituted in to 24.50 to obtain∫ b
a∥un (t)−un (s)∥p
W ds≤