1468 CHAPTER 43. INTERPOLATION IN BANACH SPACE

where a′i ∈ A′1 ∩A′0, the convergence taking place in A′1 +A′0. Now let a ∈ A0 ∩A1. FromLemma 43.8.1 ∣∣a′ (a)∣∣ ≤ ∞

∑i=−∞

∣∣a′i (a)∣∣≤

∞

∑i=−∞

J(2−i,a′i,A

′0,A′1)

K(2i,a,A0,A1

)=

∞

∑i=−∞

2−iJ(2i,a′i,A

′1,A′0)

K(2i,a,A0,A1

)≤

(∑

i

(2−(1−θ)iJ

(2i,a′i,A

′1,A′0))q′

)1/q′

·

(∑

i

(2−θ iK

(2i,a,A0,A1

))q)1/q

≤ C[∫

∞

0

(t−(1−θ)J

(t,u∗ (t) ,A′1,A

′0))q′ dt

t

]1/q′

·[∫∞

0

(t−θ K (t,a,A0,A1)

)q dtt

]1/q

.

In going from the sums to the integrals, express the first sum as a sum of integrals on[2i,2i+1) and the second sum as a sum of integrals on (2i−1,2i].

Taking the infimum over all u∗ representing a′,∣∣a′ (a)∣∣≤C∣∣∣∣a′∣∣∣∣(A′1,A

′0)1−θ ,q′,J

||a||θ ,q .

It follows a′ ∈ (A0,A1)′θ ,q and ||a′||(A0,A1)

′θ ,q≤C ||a′||(A′1,A

′0)1−θ ,q′,J

which proves the lemma.

With these two lemmas the main result follows.

Theorem 43.8.6 Suppose A0∩A1 is dense in Ai and Ai is reflexive. Then(A′1,A

′0)

1−θ ,q′ = (A0,A1)′θ ,q

and the norms are equivalent.

Proof: By Theorem 43.7.5, and the last two lemmas,

(A0,A1)′θ ,q = (A0,A1)

′θ ,q,J ⊆

(A′1,A

′0)

1−θ ,q′

=(A′1,A

′0)

1−θ ,q′,J ⊆ (A0,A1)′θ ,q .

This proves the theorem.