68.3. A MULTIPLE INTEGRAL 2339

Consider ∪p≤n{

Ip ( f ) : f ∈ Ep}

. This is a subset of P0n and so it is a subset of⊕n

i=0Hi.Now for h ∈ L2 (T n) , it was shown above that there exists a sequence gk → h in L2 (T n)where each hk ∈ En. Then In (gk)→ In (h). In particular, if h ∈ L2 (T )≡ H, then there is asequence gk ∈ E1 such that gk→ h in L2 (T ) . Then clearly

E(|W (gk)−W (h)|2

)= E

(|W (gk−h)|2

)= ∥gk−h∥2

L2(T )→ 0

and so each polynomial p(W (h1) , · · · ,W (hk)) can be approximated in L2 (Ω) by onewhich is of the form p(W (g1) , · · · ,W (gk)) where each g j ∈ E1. Corresponding to eachg j there is a list of disjoint sets. Now consider the union of all the sets just described andlet {Ak} be a partition of this union such that the Ak are pairwise disjoint and for each j,every set corresponding to g j is partitioned by a subset of the {Ak}. Thus

g j = ∑i

ciXBi = ∑i

ci

m j

∑s=1

XAis

where Bi is partitioned by the Ais. Then consider p(g1, · · · ,gk) . Then the terms of degree m

are of the formpm ≡∑

iciXAi1×···×Aim

(68.3.21)

where the Aik come from the list of disjoint sets {Ak}. The terms of degree m in

p(W (g1) , · · · ,W (gk))

are also of the form

pm (W (g1) , · · · ,W (gk))≡∑i

ci ∏k

W(Aik

)The problem is that 68.3.21 is not in Em because it is not known whether ci = 0 if

two indices are repeated. However, as explained in the proof of Lemma 68.3.11 there is afurther partition such that the contribution of those terms corresponding to i in which twoindices are repeated can be made as small as desired. Therefore, the terms of order m areapproximated in L2 (T m) by gm ∈ Em. Assume this approximation is good enough that,from the estimates given above in Lemma 68.3.9,

E(|Im (gm)− pm (W (g1) , · · · ,W (gk))|2

)1/2<

ε

n+1

Thus, taking a succession of partitions if necessary,

E

∣∣∣∣∣p(W (g1) , · · · ,W (gk))−n

∑m=0

Im (gm)

∣∣∣∣∣21/2

≤n

∑m=1

E(|Im (gm)− pm (W (g1) , · · · ,W (gk))|2

)1/2<

n

∑m=1

ε

n+1< ε.

This has proved the following lemma.