69.5. SOME IMBEDDING THEOREMS 2389

From Theorem 69.5.2 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

≤ 2p−1ε (||un (t)||p + ||un (s)||p)+Cε |t− s|p/q

This is substituted in to 69.5.30 to obtain∫ b

a||(un (t)−un (s))||pW ds≤

k

∑i=1

1ti− ti−1

∫ ti

ti−1

∫ ti

ti−1

(2p−1

ε (||un (t)||p + ||un (s)||p)+Cε |t− s|p/q)

dsdt

=k

∑i=1

2pε

∫ ti

ti−1

||un (t)||pW +Cε

ti− ti−1

∫ ti

ti−1

∫ ti

ti−1

|t− s|p/q dsdt

≤ 2pε

∫ b

a||un (t)||p dt +Cε

k

∑i=1

1(ti− ti−1)

(ti− ti−1)p/q∫ ti

ti−1

∫ ti

ti−1

dsdt

= 2pε

∫ b

a||un (t)||p dt +Cε

k

∑i=1

1(ti− ti−1)

(ti− ti−1)p/q (ti− ti−1)

2

≤ 2pεRp +Cε

k

∑i=1

(ti− ti−1)1+p/q = 2p

εRp +Cε k(

b−ak

)1+p/q

.

Taking ε so small that 2pεRp < η p/8p and then choosing k sufficiently large, it follows

||un−un||Lp([a,b];W ) <η

4.

Thus k is fixed and un at a step function with k steps having values in E. Now usecompactness of the embedding of E into W to obtain a subsequence such that {un} isCauchy in Lp (a,b;W ) and use this to contradict 69.5.29. The details follow.

Suppose un (t) = ∑ki=1 un

i X[ti−1,ti) (t) . Thus

||un (t)||E =k

∑i=1||un

i ||E X[ti−1,ti) (t)

and so

R≥∫ b

a||un (t)||pE dt =

Tk

k

∑i=1||un

i ||pE

Therefore, the {uni } are all bounded. It follows that after taking subsequences k times there

exists a subsequence{

unk

}such that unk is a Cauchy sequence in Lp (a,b;W ) . You simply

get a subsequence such that unki is a Cauchy sequence in W for each i. Then denoting this

subsequence by n,

||un−um||Lp(a,b;W ) ≤ ||un−un||Lp(a,b;W )

+ ||un−um||Lp(a,b;W )+ ||um−um||Lp(a,b;W )

≤ η

4+ ||un−um||Lp(a,b;W )+

η

4< η