1446 CHAPTER 43. INTERPOLATION IN BANACH SPACE

Proof: By Proposition 7.6.5 on Page 144 it suffices to show S has an η net in thecomplete metric space Lp ([a,b] ;W ) for each η > 0.

If not, there exists η > 0 and a sequence {un} ⊆ S, such that

||un−um|| ≥ η (43.5.24)

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)≡k

∑i=1

uniX[ti−1,ti) (t) , uni ≡1

ti− ti−1

∫ ti

ti−1

un (s)ds.

The idea is to show that un approximates un well and then to argue that a subsequence ofthe {un} is a Cauchy sequence yielding a contradiction to 43.5.24.

Therefore,

un (t)−un (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 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))||pW 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. (43.5.25)

From Theorems 43.5.2 and 43.1.9, if ε > 0, there exists Cε such that

||un (t)−un (s)||pW ≤ ε ||un (t)−un (s)||pE +Cε ||un (t)−un (s)||pX