2240 CHAPTER 65. STOCHASTIC INTEGRATION
Then for x ∈ Q1/2U,
L
∑j=1
√λ je j⊗Q1/2U
J∗e j√λ j
(x) =L
∑j=1
√λ je j
(J∗e j√
λ j,x
)Q1/2(U)
=L
∑j=1
√λ je j
(e j√λ j
,Jx
)U1
=L
∑j=1
e j (e j,Jx)U1
=∞
∑j=1
e j (e j,Jx)U1= Jx.
(J(
Q1/2 (U))⊆ span(e1, · · · ,eL) if L < ∞
)Thus,
J =L
∑j=1
√λ je j⊗Q1/2U
J∗e j√λ j
It follows that an orthonormal basis in JQ1/2U is{
JJ∗e j√λ j
}L
j=1. This is because an or-
thonormal basis for Q1/2U is{
J∗ek√λ k
}. Since J is one to one, it preserves norms between
Q1/2U and JQ1/2U . Let Φ ∈L2(JQ1/2U,H
). Then by the discussion of Hilbert Schmidt
operators given earlier, in particular the demonstration that these operators are compact,
Φ =∞
∑i=1
∞
∑j=1
φ i j fi⊗JQ1/2UJJ∗e j√
λ j
where { fi} is an orthonormal basis for H. In fact,{
fi⊗JJ∗e j√
λ j
}i, j
is an orthonormal basis
for L2(JQ1/2U,H
)and ∑i ∑ j φ
2i j < ∞, the φ i j being the Fourier coefficients of Φ. Then
consider
Φn =n
∑i=1
n
∑j=1
φ i j fi⊗JQ1/2UJJ∗e j√
λ j(65.4.6)
Consider one of the finitely many operators in this sum. For x ∈ JQ1/2U, since J preservesnorms,
fi⊗JQ1/2UJJ∗e j√
λ j(x)≡ fi
(JJ∗e j√
λ j,x
)JQ1/2U
= fi
(J∗e j√
λ j,J−1x
)Q1/2U
= fi
(e j√λ j
,JJ−1x
)U1
= fi
(e j√λ j
,x
)U1
≡ Λi j (x)
Recall how, since J is one to one, it preserves norms and inner products. Now Λi j makessense from the above formula for all x ∈U1 and is also a continuous linear map from U1 to