74.5. CONVERGENCE 2511

Also, restoring the superscript to identify the parition,

B(

X(

tk1

)−X0k

)= B(X0−X0k)+

∫ tk1

0Y (s)ds+BM

(tk1

).

Of course ∥X−X0k∥K is not bounded, but for each k it is finite. There is a sequence of par-titions Pk,∥Pk∥→ 0 such that all the above holds. In the definitions of K,K′,E ([M] (T ))replace [0,T ] with [0, t] and let the resulting spaces be denoted by Kt ,K′t . Let nk denote asubsequence of {k} such that

∥X−X0k∥Ktnk1

< 1/k.

Then from the above lemma,

E(⟨

B(X(tnk1

)−X0k

),X(tnk1

)−X0k

⟩)≤C

(⟨B(X0−X0k) ,X0−X0k⟩L1(Ω) , ||Y ||K′

tnk1

,∥X−X0k∥Ktnk1

,E([M](tnk1

)))(74.4.18)

≤C

(⟨B(X0−X0k) ,X0−X0k⟩L1(Ω) , ||Y ||K′

tnk1

,1k,E([M](tnk1

)))Hence

E(⟨

B(X(tnk1

)−X0

),X(tnk1

)−X0

⟩)≤ 2E

(⟨B(X(tnk1

)−X0k

),X(tnk1

)−X0k

⟩)+2E (⟨B(X0k−X0) ,X0k−X0⟩)

≤ 2C

(⟨B(X0−X0k) ,X0−X0k⟩L1(Ω) , ||Y ||K′

tnk1

,1k,E([M](tnk1

)))+2∥B∥∥X0k−X0∥2

L2(Ω,W )

which converges to 0 as k→ ∞. It follows that there exists a suitable subsequence suchthat 74.4.17 holds even in the case that X0 is only known to be in L2 (Ω,W ). From now on,assume this subsequence for the partitions Pk. Thus k will really be nk and it suffices toconsider the limit as k→ ∞ of the equation of 74.4.17. To emphasize this point again, thereason for the above observations is to argue that, even when X0 is only in L2 (Ω,W ) , onecan neglect

⟨B(X (t1)−X0−M (t1)) ,X (t1)−X0−M (t1)⟩in passing to the limit as k→ ∞ provided a suitable subsequence is used.

74.5 ConvergenceConvergence will be shown for a subsequence and from now on every sequence will bea subsequence of this one. Since BX ∈ L2 ([0,T ]×Ω;W ′) which was shown above, thereexists a sequence of partitions of the sort described above such that also, in addition to theother claims

BX lk → BX ,BX r

k → BX

in L2 ([0,T ]×Ω,W ′). Then the next lemma improves on this.