72.4. THE MAIN ESTIMATE 2449

≤C+C∫

Ω

(∫ T

0||Z (u)||2

∣∣∣X lk (u)

∣∣∣2 du)1/2

dP

Now for each ω, there are only finitely many values of X lk (u) and they equal X (t j) for

t j ∈Pk with the convention that X (0) = 0. Therefore, the above is dominated by

C+C∫

Ω

(sup

t j∈Pk

∣∣X (t j)∣∣2)1/2(∫ T

0||Z (u)||2 du

)1/2

dP

≤C+12

∫Ω

supt j∈Pk

∣∣X (t j)∣∣2 +C

∫Ω

∫ T

0||Z (u)||2

L2(Q1/2U,H) dudP

and so12

∫Ω

supt j∈Pk

|X (tk)|2 dP≤C

for some constant C independent of Pk dependent on∫

Ω

∫ T0 ||Z (u)||2

L2(Q1/2U,H) dudP. This

constant is dependent on ||Y ||K′ , ||X̄ ||K , ||Z||J ,∥X0∥L2(Ω,H) and equals zero when all ofthese quantities equal 0.

Let D denote the union of all the Pk. Thus D is a dense subset of [0,T ] and it has justbeen shown that for a constant C independent of Pk,

E(

supt∈D|X (t)|2

)≤C.

Let{

e j}

be an orthonormal basis for H which is also contained in V and has the prop-erty that span({ek}∞

k=1) is dense in V . I claim that for y ∈V ′

|y|2H = supn

n

∑j=1

∣∣⟨y,e j⟩∣∣2

This is certainly true if y ∈ H because in this case⟨y,e j

⟩= (y,e j)

If y /∈ H, then the series must diverge. If not, you could consider the infinite sum

z≡∞

∑j=1

⟨y,e j

⟩e j ∈ H

and argue that ⟨z− y,v⟩= 0 for all v ∈ span({ek}∞

k=1) which would also imply that this istrue for all v ∈V . Then since z = y in V ′, it follows that y ∈ H contrary to the assumptionthat y /∈ H.

It follows

|X (t)|2 = supn

n

∑j=1

∣∣⟨X (t) ,e j⟩∣∣2