1466 CHAPTER 43. INTERPOLATION IN BANACH SPACE

= ∑i

[2−(i−1)θ J

(2i,bi

)α i

J (2i,bi) ln2

]q

ln2

=C∑i

(2−iθ |α i|

)q< ∞ (43.8.64)

and so ||a∞||θ ,q,J < ∞. Now for a′ as above, a′ ∈ (A0,A1)′θ ,q,J ⊆ (A0 +A1)

′ , and so sincethe sum for a∞ converges in A0 +A1, we have

a′ (a∞) = ∑i

J(2i,bi

)−1α ia′ (bi) .

Therefore,

a′ (a∞) ≥ ∑i

[K(2−i,a′

)− ε min

(1,2−i)]

α i

= ∑i

K(2−i,a′

)α i−∑

iε min

(1,2−i)

α i

= ∑i

K(2−i,a′

)α i−O(ε) (43.8.65)

The reason for this is that α ∈ λθ ,q so

{α i2−iθ

}∈ lq. Therefore,

∑i

ε min(1,2−i)

α i = ε

{∞

∑i=0

2−iα i +

−1

∑i=−∞

α i

}

= ε

{∞

∑i=0

2−iθ 2(θ−1)iα i +

−1

∑i=−∞

α i2−iθ 2iθ

}

≤ ε

(

∑i

∣∣∣α i2−iθ∣∣∣q)1/q(

∞

∑i=0

(2(θ−1)i

)q′)1/q′

+

(∑

i

∣∣∣α i2−iθ∣∣∣q)1/q(

∞

∑i=0

(2θ i)q′)1/q′

<Cε

Also |a′ (a∞)| ≤ ||a′||(A0,A1)′θ ,q,J||a∞||(A0,A1)θ ,q,J

.Now from the definition of K,

K(2−i,a′,A′0,A

′1)= 2−iK

(2i,a′,A′1,A

′0)

and so from 43.8.65

∑i

2−iK(2i,a′,A′1,A

′0)

α i−O(ε) ≤ a′ (a∞)

≤∣∣∣∣a′∣∣∣∣

(A0,A1)′θ ,q,J

Cθ ||α||λ θ ,q .