43.3. THE IMPLICIT CASE 1435

equaling 0 by 43.3.12. From 43.3.15 and 43.3.13,

≤ ε +∥B∥1/2

(∞

∑i=1|⟨B(xk− x) ,ei⟩|2

)1/2

≤ ε +∥B∥1/2 ⟨B(xk− x) ,xk− x⟩1/2 < 2ε

whenever k is large enough. Therefore,

Bx =∞

∑i=1⟨Bx,ei⟩Bei

in W ′. It follows that

⟨Bx,x⟩= limk→∞

⟨k

∑i=1⟨Bx,ei⟩Bei,x

⟩= lim

k→∞

k

∑i=1|⟨Bx,ei⟩|2 ≡

∞

∑i=1|⟨Bx,ei⟩|2

Theorem 43.3.2 Let V ⊆W,W ′ ⊆ V ′ be separable Banach spaces,W a separable Hilbertspace, and let Y ∈ Lp′ (0,T ;V ′)≡ K′ and

BX (t) = BX0 +∫ t

0Y (s)ds in V ′ (43.3.16)

where X0 ∈W, and it is known that X ∈ Lp (0,T,V ) ≡ K for p > 1. Also assume X ∈L2 (0,T,W ) . Then t→ BX (t) is in C ([0,T ] ,W ′) and also

12⟨BX (t) ,X (t)⟩= 1

2⟨BX0,X0⟩+

∫ t

0⟨Y (s) ,X (s)⟩ds

Proof: By Lemma 43.2.1, 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 also BX l ,BX r→ BX in L2 ([0,T ] ,W ′).

Lemma 43.3.3 Let s < t. Then for X ,Y satisfying 43.3.16

⟨BX (t) ,X (t)⟩= ⟨BX (s) ,X (s)⟩

+2∫ t

s⟨Y (u) ,X (t)⟩du−⟨B(X (t)−X (s)) ,(X (t)−X (s))⟩ (43.3.17)