556 CHAPTER 19. HILBERT SPACES

T such that limn→∞ ∥Tn−T∥ = 0. Then it only remains to verify T ∈L2 (H,G) . But byFatou’s lemma,

∑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) asclaimed.

∑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). Next I show they are orthonormal.

( 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.

Now 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

It follows ∥Tnek∥ ≤ ∥Tek∥ and limn→∞ Tnek = Tek. Therefore, from the dominated conver-gence 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.

This also shows L2 (H,G) is separable. From 19.8.64 it also shows that every T ∈L2 (H,G) is the limit in the operator norm of a sequence of compact operators. Thisfollows because each of the f j⊗ei is easily seen to be a compact operator because if B⊆His bounded, then ( f j⊗ ei)(B) is a bounded subset of a one dimensional vector space so it ispre-compact. Thus Tn is compact, being a finite sum of these. By Lemma 19.8.6, so is T .

Finally, consider 19.8.65.

∥X⊗Y∥2L2≡∑

k|X⊗Y ( fk)|2H ≡∑

k|X ( fk,Y )|2H

= ∥X∥2H ∑

k|( fk,Y )|2 = ∥X∥2

H ∥Y∥2G