79.1. THE CASE OF UNIQUENESS 2695

N, Bu(T,ω) = B(u(T,ω)) , similar for Bur. If not, just do the following argument forall T ′ close to T , letting T ′ be in the dense subset just described. Then from the inte-gral equation solved, and letting {ei} be the special set described in Theorem 77.2.19 andsuppressing the dependence on ω,

∞

∑i=1⟨Bur (T ) ,ei⟩2−

∞

∑i=1⟨Bu0,ei⟩+2

∫ T

0⟨zr,ur⟩ds

= 2∫ T

0⟨ f ,ur⟩ds+2

∫ T

0

(Φr ◦ J−1)∗Bur ◦ JdW

Thus also

2∫ T

0⟨zr,ur⟩ds =−

∞

∑i=1⟨Bur (T ) ,ei⟩2 +

∞

∑i=1⟨Bu0,ei⟩

+2∫ T

0⟨ f ,ur⟩ds+2

∫ T

0

(Φr ◦ J−1)∗Bur ◦ JdW (79.1.31)

A similar formula to 79.1.31 holds for u. Thus

2∫ T

0⟨z,u⟩ds =−

∞

∑i=1⟨Bu(T ) ,ei⟩2 +

∞

∑i=1⟨Bu0,ei⟩

+2∫ T

0⟨ f ,u⟩ds+2

∫ T

0

(Φ◦ J−1)∗Bu◦ JdW

It follows from 79.1.25 and the other convergences that

lim supr→∞

∫ T

0⟨zr,ur⟩ds≤

∫ T

0⟨z,u⟩ds

Hencelim sup

r→∞

⟨zr,ur−u⟩V ′,V ≤ 0

Now from the limit condition, for any v ∈ V , there exists a z(v) ∈ A(u(·,ω) ,ω) such that

⟨z,u− v⟩V ′,V ≥ lim infr→∞

(⟨zr,ur−u⟩+ ⟨zr,u− v⟩)

≥ lim infr→∞⟨zr,ur− v⟩ ≥ ⟨z(v) ,u− v⟩

The reason the limit condition applies is the estimate 79.1.29 and the convergence 79.1.24which shows that

B(

ur−∫ (·)

0ΦrdW

)satisfy a Holder condition into V ′. Then the estimate 79.1.29 implies that the B

∫ (·)0 ΦrdW

are bounded in a Holder norm and so the same is true of the Bur. Thus the situation of thelimit condition 79.1.7 is obtained. Then it follows from separation theorems and the factthat A(u(·,ω) ,ω) is closed and convex that z(·,ω) ∈ A(u(·,ω) ,ω). This has proved thefollowing Theorem.