61.9. ABSTRACT WIENER SPACES 2033

Lemma 61.9.6 The above definition is well defined.

Proof: Let { fk} be another orthonormal set such that for F,G Borel sets in Rn,

A = {x ∈ H : ((x,e1) , · · · ,(x,en)) ∈ F}= {x ∈ H : ((x, f1) , · · · ,(x, fn)) ∈ G}

I need to verify ν (A) is the same using either { fk} or {ek}. Let a ∈ G. Then

x≡n

∑i=1

ai fi ∈ A

because (x, fk) = ak. Therefore, for this x it is also true that ((x,e1) , · · ·(x,en))∈ F. In otherwords for a ∈ G, (

n

∑i=1

(e1, fi)ai, · · · ,n

∑i=1

(en, fi)ai

)∈ F

Let L ∈L (Rn,Rn) be defined by

La≡∑i

L jiai, L ji ≡ (e j, fi) .

Since the{

e j}

and { fk} are orthonormal, this mapping is unitary. Also this has shown that

LG⊆ F.

SimilarlyL∗F ⊆ G

where L∗ has the i jth entry L∗i j = ( fi,e j) as above and L∗ is the inverse of L because L isunitary. Thus

F = L(L∗ (F))⊆ L(G)⊆ F

showing that LG = F and L∗F = G.Now let θ et≡∑i tiei with θ f defined similarly. Then the definition of ν (A) correspond-

ing to {ei} is

ν (A)≡ 1

(2π)n/2 (det(θ ∗eQθ e))1/2

∫F

e−12 t∗θ∗eQ−1θ etdt

Now change the variables letting t = Ls where s ∈ G.From the definition,

θ eLs = ∑j∑

i(e j, fi)sie j

= ∑j

(e j,∑

ifisi

)e j = ∑

j

(e j,θ f s

)e j

and soLs =

((e1,θ f s

), · · · ,

(en,θ f s

))= θ

∗eθ f s