69.3. AN IMPORTANT FORMULA 2369

Lemma 69.3.1 Let Y : [0,T ]→ E, be B ([0,T ]) measurable and suppose

Y ∈ Lp (0,T ;E)≡ K, p≥ 1

Then there exists a sequence of nested partitions, Pk ⊆Pk+1,

Pk ≡{

tk0 , · · · , tk

mk

}such that the step functions given by

Y rk (t) ≡

mk

∑j=1

Y(

tkj

)X[tk

j−1,tkj )(t)

Y lk (t) ≡

mk

∑j=1

Y(

tkj−1

)X(tk

j−1,tkj ](t)

both converge to Y in K as k→ ∞ and

limk→∞

max{∣∣∣tk

j − tkj+1

∣∣∣ : j ∈ {0, · · · ,mk}}= 0.

Also, each Y(

tkj

),Y(

tkj−1

)is in E. One can also assume that Y (0) = 0. The mesh points{

tkj

}mk

j=0can be chosen to miss a given set of measure zero. In addition to this, we can

assume that ∣∣∣tkj − tk

j−1

∣∣∣= 2−nk

except for the case where j = 1 or j = mnk when this might not be so. In the case of the lastsubinterval defined by the partition, we can assume∣∣∣tk

m− tkm−1

∣∣∣= ∣∣∣T − tkm−1

∣∣∣≥ 2−(nk+1)

Theorem 69.3.2 Let V ⊆H =H ′⊆V ′ be a Gelfand triple and suppose Y ∈ Lp′ (0,T ;V ′)≡K′ and

X (t) = X0 +∫ t

0Y (s)ds in V ′ (69.3.15)

where X0 ∈ H, and it is known that X ∈ Lp (0,T,V ) ≡ K for p > 1. Then t → X (t) is inC ([0,T ] ,H) and also

12|X (t)|2H =

12|X0|2H +

∫ t

0⟨Y (s) ,X (s)⟩ds

Proof: By Lemma 65.3.1, there exists a sequence of uniform partitions{

tnk

}mnk=0 =

Pn,Pn ⊆Pn+1, of [0,T ] such that the step functions

mn−1

∑k=0

X (tnk )X(tn

k ,tnk+1]

(t) ≡ X l (t)

mn−1

∑k=0

X(tnk+1)X(tn

k ,tnk+1]

(t) ≡ X r (t)

converge to X in K and in L2 ([0,T ] ,H).