76.4. THE GENERAL CASE 2555
Recall thatV ⊆W, W ′ ⊆V ′
each space dense in the one to its right and the inclusion maps are continuous.Assume only
Φ ∈ L2([0,T ]×Ω,L2
(Q1/2U,W
)).
By density of E into W, there exists a sequence
Φn ∈ L2([0,T ]×Ω,L2
(Q1/2U,E
))such that
∥Φn−Φ∥L2([0,T ]×Ω,L2(Q1/2U,W))→ 0,
∥Φn∥L2(Q1/2U,W) ≤ ∥Φ∥L2(Q1/2U,W) .
Also let u0n ∈ Lp (Ω,V ) where u0n is F0 measurable and such that u0n ∈ Lp (Ω,V ) and
∥u0n (ω)−u0 (ω)∥W → 0, ⟨Bu0n,u0n⟩ ≤ 2⟨Bu0,u0⟩
for each ω . The existence of such an approximating sequence follows from density con-siderations of E into V and of V into W .
By Lemma 76.3.8 there is a solution un to the integral equation
Bun (t)−Bu0n +∫ t
0A(s,un,ω)ds =
∫ t
0f (s,ω)ds+B
∫ t
0ΦndW (76.4.27)
Then by the Implicit Ito formula there is a set of measure zero such that for all n,m
12⟨B(un−um) ,un−um⟩(t)−
12⟨Bu0n−Bu0m,u0n−u0m⟩
+δ
∫ t
0∥un−um∥α
U ds
≤ λ
∫ t
0⟨B(un−um) ,un−um⟩(s)ds
+12
∫ t
0⟨B(Φn−Φm) ,Φn−Φm⟩L2
ds+Mmn (t)
Also the last term is a martingale whose quadratic variation satisfies
[Mmn] (t)≤C∫ t
0∥Φn−Φm∥2
L2(Q1/2U,W) ∥B(un−um)∥2W ′ ds
≤C∫ t
0∥Φn−Φm∥2
L2(Q1/2U,W) ⟨Bun−Bum,un−um⟩ds