34.5. THE IMPLICIT CASE, B = B(t) 1205

Hence {Bu(t (k))}∞

k=1 is a convergent sequence in W ′ because

|⟨B(u(t (k))−u(t (m))) ,y⟩|≤ ⟨B(u(t (k))−u(t (m))) ,u(t (k))−u(t (m))⟩1/2 ⟨By,y⟩1/2

≤ ⟨B(u(t (k))−u(t (m))) ,u(t (k))−u(t (m))⟩1/2 ∥B∥1/2 ∥y∥W

Does it converge to Bu(t)? Let ξ (t) ∈W ′ be what it does converge to. Let v ∈V. Then

⟨ξ (t) ,v⟩= limk→∞

⟨Bu(t (k)) ,v⟩= limk→∞

⟨Bu(t (k)) ,v⟩= ⟨Bu(t) ,v⟩

because it is known that t → Bu(t) is continuous into V ′. It is also known that for t ∈ NC,Bu(t) ∈W ′ ⊆ V ′ and that the Bu(t) for t ∈ NC are uniformly bounded in W ′. Therefore,since V is dense in W, it follows that ξ (t) = Bu(t).

Now for every t ∈ D, it was shown above that

⟨Bu(t) ,u(t)⟩+∫ t

0

⟨B′u,u

⟩dr = ⟨Bu0,u0⟩+2

∫ t

0⟨Y (r) ,u(r)⟩dr

Also it was just shown that Bu(t (k))→ Bu(t) for t /∈ N. Then for t /∈ N

|⟨Bu(t (k)) ,u(t (k))⟩−⟨Bu(t) ,u(t)⟩|

≤ |⟨B(t (k))u(t (k)) ,u(t (k))−u(t)⟩|+ |⟨Bu(t (k))−Bu(t) ,u(t)⟩|

Then the second term converges to 0. The first equals

|⟨B(t (k))u(t (k))−B(t (k))u(t) ,u(t (k))⟩|≤ ⟨B(t (k))(u(t (k))−u(t)) ,u(t (k))−u(t)⟩1/2 ⟨Bu(t (k)) ,u(t (k))⟩1/2

From the above, this is dominated by an expression of the form

⟨B(t (k))(u(t (k))−u(t)) ,u(t (k))−u(t)⟩1/2 C

Then from the choice of N and the pointwise convergence of urn to u off N the above

converges to 0 for each t /∈ N. It follows that

limk→∞

|⟨Bu(t (k)) ,u(t (k))⟩−⟨Bu(t) ,u(t)⟩|= 0

Then from the formula,

⟨Bu(t) ,u(t)⟩= ⟨Bu0,u0⟩+2∫ t

0⟨Y (r) ,u(r)⟩dr−

∫ t

0

⟨B′u,u

⟩dr

valid for t ∈ D, it follows that the same formula holds for all t /∈ N. Then define ⟨Bu,u⟩(t)to equal ⟨Bu(t) ,u(t)⟩ off N and the right side for t ∈ N. Thus t→ ⟨Bu,u⟩(t) is continuousand for all t ∈ [0,T ] ,

⟨Bu,u⟩(t) = ⟨Bu0,u0⟩+2∫ t

0⟨Y (r) ,u(r)⟩dr−

∫ t

0

⟨B′u,u

⟩dr