1434 CHAPTER 43. INTERPOLATION IN BANACH SPACE

By 43.3.13,

≤

(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 (43.3.14)

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 43.3.14

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 (43.3.15)

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

∥∥∥∥∥