2710 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS
where Mn (t) is a local martingale whose quadratic variation is
[Mn] (t)≤C∫ t
0||Φ||2L2
|un|2H ds
Then estimates give∥∥ 1
n ⟨Fun,un⟩∥∥
U ′ bounded as well as
∥un∥V ,∥zn∥V ′ (79.4.46)
One takes expectation of 79.4.45 using an appropriate localizing sequence of stoppingtimes if necessary. It follows that there is a subsequence, still denoted with n such that
un → u weakly in V (79.4.47)zn → z weakly in V ′
1n
Fun → 0 strongly in U ′
The last convergence follows from the following argument.∫ T
0
∫Ω
1n⟨Fun,w⟩dPdt
≤∫ T
0
∫Ω
1n1/r′
⟨Fun,un⟩1/r′ 1n1/r⟨Fw,w⟩1/r dPdt
≤(∫ T
0
∫Ω
1n⟨Fun,un⟩dPdt
)1/r′(∫ T
0
∫Ω
1n∥w∥r dPdt
)1/r
≤ C1
n1/r∥w∥U
and so ∣∣∣∣∣∣∣∣1nFun
∣∣∣∣∣∣∣∣U ′≤ C
n1/r
Recall
un (t)−u0 +1n
∫ t
0Funds+
∫ t
0znds
=∫ t
0f ds+
∫ t
0ΦdW, zn ∈ Â(un)
Then if φ ∈U , ⟨1n
∫ (·)
0Funds+
∫ (·)
0znds,φ
⟩U ′,U
=∫
Ω
∫ T
0
⟨(1n
∫ r
0Funds+
∫ r
0znds
),φ (r,ω)
⟩U ′,U
drdP
Then the above equals∫Ω
∫ T
0
1n
∫ r
0⟨Fun (s,ω) ,φ (r,ω)⟩dsdrdP+
∫ r
0⟨zn (s,ω) ,φ (r,ω)⟩dsdrdP