2046 CHAPTER 61. PROBABILITY IN INFINITE DIMENSIONS

and the series converges in H1 because

∞

∑k=1

λ k |(x,ek)| ≤

(∞

∑k=1

λ2k

)1/2(∞

∑k=1|(x,ek)|2

)1/2

< ∞.

Also H is a Hilbert space with inner product given by

(x,y)H ≡(A−1x,A−1y

)H1

.

H is complete because if {xn} is a Cauchy sequence in H, this is the same as{

A−1xn}

being a Cauchy sequence in H1 which implies A−1xn→ y for some y ∈H1. Then it followsxn = A

(A−1xn

)→ Ay in H.

For x ∈ H ⊆ H1,

||x|| ≤ |x|H1=∣∣AA−1x

∣∣H1≤ ||A||

∣∣A−1x∣∣H1≡ ||A|| |x|H

and so the embedding of H into E is continuous. Why is ||·|| a measurable norm on H?Note first that for x ∈ H ⊆ H1,

|Ax|H ≡∣∣A−1Ax

∣∣H1

= |x|H1≥ ||x||E . (61.11.49)

Therefore, if it can be shown A is a Hilbert Schmidt operator on H, the desired measurabil-ity will follow from Lemma 61.9.11 on Page 2037.

Claim: A is a Hilbert Schmidt operator on H.

Proof of the claim: From the definition of the inner product in H, it follows an or-thonormal basis for H is {λ kek} . This is because

(λ kek,λ je j)H ≡(λ kA−1ek,λ jA−1e j

)H1

= (ek,e j)H1= δ jk.

To show that A is Hilbert Schmidt, it suffices to show that

∑k|A(λ kek)|2H < ∞

because this is the definition of an operator being Hilbert Schmidt. However, the aboveequals

∑k

∣∣A−1A(λ kek)∣∣2H1

= ∑k

λ2k < ∞.

This proves the claim.Now consider 61.11.49. By Lemma 61.9.11, it follows the norm ||x||′ ≡ |Ax|H is Gross

measurable on H. Therefore, ||·||E is also Gross measurable because it is smaller. Thisproves the theorem.

Using Theorem 61.11.3 and Theorem 61.10.1 this proves most of the following impor-tant corollary.