2378 CHAPTER 69. GELFAND TRIPLES

It follows that∞

∑i=1⟨Bx,ei⟩Bei (69.4.21)

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 69.4.21

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 (69.4.22)

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∥

+

∥∥∥∥∥Bxk−∞

∑i=1⟨Bxk,ei⟩Bei

∥∥∥∥∥+∥∥∥∥∥ ∞

∑i=1⟨Bxk,ei⟩Bei−

∞

∑i=1⟨Bx,ei⟩Bei

∥∥∥∥∥≤ ε +

∥∥∥∥∥ ∞

∑i=1⟨B(xk− x) ,ei⟩Bei

∥∥∥∥∥ ,the term ∥∥∥∥∥Bxk−

∞

∑i=1⟨Bxk,ei⟩Bei

∥∥∥∥∥equaling 0 by 69.4.19. From 69.4.22 and 69.4.20,

≤ ε +∥B∥1/2

(∞

∑i=1|⟨B(xk− x) ,ei⟩|2

)1/2

≤ ε +∥B∥1/2 ⟨B(xk− x) ,xk− x⟩1/2 < 2ε