594 CHAPTER 22. HILBERT SPACES

that limn→∞ ∥Tn−T∥ = 0. Then it only remains to verify T ∈L2 (H,G) . But by Fatou’slemma, for {ek} orthonormal,

∑k∥Tek∥2 ≤ lim inf

n→∞∑k∥Tnek∥2 ≡ lim inf

n→∞∥Tn∥2

L2< ∞.

All that remains is to verify L2 (H,G) is separable and these Hilbert Schmidt operatorsare compact. I will show an orthonormal basis for L2 (H,G) is

{f j⊗ ek

}where { fk} is

an orthonormal basis for G and {ek} is an orthonormal basis for H. Here, for f ∈ G ande ∈ H, f ⊗ e(x)≡ (x,e) f .

I need to show f j ⊗ ek ∈L2 (H,G) and that it is an orthonormal basis for L2 (H,G)as claimed. Let the {ek} be the orthonormal basis used to define the inner product but the{

f j}

are just an arbitrary orthonormal basis for G.

∑k

∥∥ f j⊗ ei (ek)∥∥2

= ∑k

∥∥ f jδ ik∥∥2

=∥∥ f j∥∥2

= 1 < ∞

so each of these operators is in L2 (H,G). As noted above, they are also each continuous.Next I show they are orthonormal. From the definition of the inner product,

( f j⊗ ek, fs⊗ er) = ∑p( f j⊗ ek (ep) , fs⊗ er (ep))

= ∑p

δ rpδ kp ( f j, fs) = ∑p

δ rpδ kpδ js

If j = s and k = r this reduces to 1. Otherwise, this gives 0. Thus these operators areorthonormal.

Why is L2 (H,G) a separable Hilbert space? Let T ∈L2 (H,G). Consider

Tn ≡n

∑i=1

n

∑j=1

(Tei, f j) f j⊗ ei

Then

Tnek =n

∑i=1

n

∑j=1

(Tei, f j)(ek,ei) f j =n

∑j=1

(Tek, f j) f j,

a partial sum for Tek. It follows ∥Tnek∥ ≤ ∥Tek∥ and limn→∞ Tnek = Tek. Therefore, fromthe dominated convergence theorem,

limn→∞∥T −Tn∥2

L2≡ lim

n→∞∑k∥(T −Tn)ek∥2 = 0.

Therefore, the linear combinations of the f j ⊗ ei are dense in L2 (H,G) and this provescompleteness of the orthonomal basis.

By only using rational scalars in the linear combinations we see that L2 (H,G) is sep-arable. From 22.20 it also shows that every T ∈L2 (H,G) is the limit in the operator normof a sequence of compact operators. This follows because each of the f j⊗ ei is easily seento be a compact operator because if B⊆H is bounded, then ( f j⊗ ei)(B) is a bounded sub-set of a one dimensional vector space so it is pre-compact. Thus Tn is compact, being afinite sum of these. By Lemma 22.5.10, so is T .