34.4. THE IMPLICIT CASE 1183
Thus the series on the left converges. Then also, from the above inequality,∣∣∣∣∣⟨
q
∑i=p⟨Bx,ei⟩Bei,y
⟩∣∣∣∣∣≤ q
∑i=p|⟨Bx,ei⟩| |⟨Bei,y⟩|
≤
(q
∑i=p|⟨Bx,ei⟩|2
)1/2( q
∑i=p|⟨By,ei⟩|2
)1/2
≤
(q
∑i=p|⟨Bx,ei⟩|2
)1/2(∞
∑i=1|⟨By,ei⟩|2
)1/2
By 34.4.19,
≤
(q
∑i=p|⟨Bx,ei⟩|2
)1/2(∥B∥∥y∥2
W
)1/2≤
(q
∑i=p|⟨Bx,ei⟩|2
)1/2
∥B∥1/2 ∥y∥W
It follows that∞
∑i=1⟨Bx,ei⟩Bei (34.4.20)
converges in W ′ because it was just shown that∥∥∥∥∥ q
∑i=p⟨Bx,ei⟩Bei
∥∥∥∥∥W ′≤
(q
∑i=p|⟨Bx,ei⟩|2
)1/2
∥B∥1/2
and it was shown above that ∑∞i=1 |⟨Bx,ei⟩|2 < ∞, so the partial sums of the series 34.4.20
are a Cauchy sequence in W ′. Also, the above estimate shows that for ∥y∥= 1,∣∣∣∣∣⟨
∞
∑i=1⟨Bx,ei⟩Bei,y
⟩∣∣∣∣∣ ≤(
∞
∑i=1|⟨By,ei⟩|2
)1/2(∞
∑i=1|⟨Bx,ei⟩|2
)1/2
≤
(∞
∑i=1|⟨Bx,ei⟩|2
)1/2
∥B∥1/2
and so ∥∥∥∥∥ ∞
∑i=1⟨Bx,ei⟩Bei
∥∥∥∥∥W ′≤
(∞
∑i=1|⟨Bx,ei⟩|2
)1/2
∥B∥1/2 (34.4.21)
Now for x arbitrary, let xk ∈ span({
g j}∞
j=1
)and xk→ x in W. Then for a fixed k large
enough, ∥∥∥∥∥Bx−∞
∑i=1⟨Bx,ei⟩Bei
∥∥∥∥∥≤ ∥Bx−Bxk∥