65.5. THE GENERAL INTEGRAL 2243

Proof: First, why is Φ◦ J−1 ∈ L2([a,T ]×Ω;L2

(JQ1/2U,H

))? This follows from the

observation that A is Hilbert Schmidt if and only if A∗ is Hilbert Schmidt. In fact, the HilbertSchmidt norms of A and A∗ are the same. Now since Φ is Hilbert Schmidt, it follows thatΦ∗ is and since J−1 is continuous, it follows

(J−1)∗

Φ∗ =(Φ◦ J−1

)∗ is Hilbert Schmidt.Also letting L2 be the appropriate space of Hilbert Schmidt operators,∣∣∣∣∣∣(J−1)∗∣∣∣∣∣∣ ||Φ||L2

=∣∣∣∣∣∣(J−1)∗∣∣∣∣∣∣ ||Φ∗||L2

≥∣∣∣∣∣∣(Φ◦ J−1)∗∣∣∣∣∣∣

L2=∣∣∣∣Φ◦ J−1∣∣∣∣

L2

Thus Φ◦ J−1 has values in L2(JQ1/2U,H

). This also shows that

Φ◦ J−1 ∈ L2([a,T ]×Ω;L2

(JQ1/2U,H

)).

Since Φ is given to be progressively measurable, so is Φ◦ J−1. Therefore, the existence ofthe desired sequence of elementary functions follows from Proposition 65.3.2 and Lemma65.4.5.

Definition 65.5.2 Let Φ ∈ L2([a,T ]×Ω;L2

(Q1/2U,H

))and be progressively measur-

able where Q is a self adjoint nonnegative operator defined on U. Let J : Q1/2U →U1 beHilbert Schmidt. Then the stochastic integral∫ t

aΦdW (65.5.7)

is defined as

limn→∞

∫ t

aΦndW in L2 (Ω;H)

where W (t) is a Wiener process

∞

∑k=1

ψk (t)Jgk, {gk} orthonormal basis in Q1/2U,

and Φn is an elementary function which has values in L (U1,H) and converges to Φ◦ J−1

inL2([a,T ]×Ω;L2

(JQ1/2U,H

)),

such a sequence exists by Lemma 65.4.5 and Proposition 65.3.2.

U↓ Q1/2

U1 ⊇ JQ1/2U J←1−1

Q1/2U

Φn ↘ ↓ Φ

H

It is necessary to show that this is well defined and does not depend on the choice of U1and J.