1462 CHAPTER 43. INTERPOLATION IN BANACH SPACE

and for all a ∈ A0∩A1,

a′0 (a)+a′1 (a) = λ ((a,a)) = a′ (a) .

It follows that a′0 +a′1 = a′ in (A0∩A1)′. Therefore, from 43.8.56,

||λ ||E ′ ≤ inf{∣∣∣∣ã′0∣∣∣∣A0

+ t∣∣∣∣ã′1∣∣∣∣A1

: a′ = ã′0 + ã′1}≡ K

(t,a′)

(43.8.57)

≤∣∣∣∣a′0∣∣∣∣A′0 + t

∣∣∣∣a′1∣∣∣∣A′1 = ||λ ||E ′ ≡ supa∈A0∩A1

|a′ (a)|J (t−1,a)

(43.8.58)

because on E, J(t−1,a

)= ||(a,a)||A0×A1

which proves 43.8.52.To obtain 43.8.53 in the case that Ai is reflexive, apply 43.8.52 to the case where A′′i

plays the role of Ai in 43.8.52. Thus, for a′′ ∈ A′′0 +A′′1 ,

K(t,a′′

)= sup

a′∈A′0∩A′1

|a′′ (a′)|J (t−1,a′)

.

Now a′′ = a′′1 + a′′0 = η1a1 +η0a0 where η i is the map from Ai to A′′i which is onto andpreserves norms, given by ηa(a′)≡ a′ (a) . Therefore, letting a1 +a0 = a

K (t,a) = K(t,a′′

)= sup

a′∈A′0∩A′1

|a′′ (a′)|J (t−1,a′)

= supa′∈A′0∩A′1

|(η1a1 +η0a0)(a′)|J (t−1,a′)

= supa′∈A′0∩A′1

|(a′ (a1 +a0))|J (t−1,a′)

and so

K (t,a) = supa′∈A′0∩A′1

|a′ (a)|J (t−1,a′)

Changing t→ t−1,K(t−1,a

)J(t,a′)≥∣∣a′ (a)∣∣ .

which proves the lemma.Consider (A0,A1)

′θ ,q .

Definition 43.8.2 Let q≥ 1. Then λθ ,q will denote the sequences, {α i}∞

i=−∞such that

∞

∑i=−∞

(|α i|2−iθ

)q< ∞.

For α ∈ λθ ,q,

||α||λ

θ ,q ≡

(∞

∑i=−∞

(|α i|2−iθ

)q)1/q

.

Thus α ∈ λθ ,q means

{α i2−iθ

}∈ lq.