552 CHAPTER 19. HILBERT SPACES
the series converging because
x =∞
∑j=1
(x,e j)e j
Then also since A is self adjoint,∞
∑j=1
∞
∑i=1
(Aei,e j)ei⊗ e j (x) ≡∞
∑j=1
∞
∑i=1
(Aei,e j)(x,e j)ei
=∞
∑j=1
(x,e j)∞
∑i=1
(Aei,e j)ei
=∞
∑j=1
(x,e j)∞
∑i=1
(Ae j,ei)ei
=∞
∑j=1
(x,e j)Ae j
Next consider the claim that A is compact. Let CA ≡(
∑∞j=1∣∣(Ae j,e j)
∣∣)1/2. Let An be
defined by
An ≡∞
∑j=1
n
∑i=1
(Aei,e j)(ei⊗ e j) .
Then An has values in span(e1, · · · ,en) and so it must be a compact operator becausebounded sets in a finite dimensional space must be precompact. Then
|(Ax−Anx,y)| =
∣∣∣∣∣ ∞
∑j=1
∞
∑i=n+1
(Aeie j)(y,e j)(ei,x)
∣∣∣∣∣=
∣∣∣∣∣ ∞
∑j=1
(y,e j)∞
∑i=n+1
(Aeie j)(ei,x)
∣∣∣∣∣≤
∣∣∣∣∣ ∞
∑j=1
∣∣(y,e j)∣∣(Ae j,e j)
1/2∞
∑i=n+1
(Aeiei)1/2 |(ei,x)|
∣∣∣∣∣≤
(∞
∑j=1
∣∣(y,e j)∣∣2)1/2(
∞
∑j=1
∣∣(Ae j,e j)∣∣)1/2
·
(∞
∑i=n+1
|(x,ei)|2)1/2(
∞
∑i=n+1
|(Aeiei)|)1/2
≤ |y| |x|CA
(∞
∑i=n+1
|(Aei,ei)|)1/2
and this shows that if n is sufficiently large,
|((A−An)x,y)| ≤ ε |x| |y| .