1450 CHAPTER 43. INTERPOLATION IN BANACH SPACE

It follows there exist xn ∈ A0 and yn ∈ A1 such that xn + yn = 0 for every n and wheneverm,n are large enough,

||a0n + xn− (a0m + xm)||0 + ||a1n + yn− (a1m + ym)||1 < ε

Hence {a1n + yn} is a Cauchy sequence in A1 and {a0n + xn} is a Cauchy sequence in A0.Let

a0n + xn → a0 ∈ A0

a1n + yn → a1 ∈ A1.

Then

K (1,a0n +a1n− (a0 +a1)) = K (1,a0n + xn +a1n + yn− (a0 +a1))

≤ ||a0n + xn−a0||0 + ||a1n + yn−a1||1

which converges to 0. Thus A0 +A1 is a Banach space as claimed.With this, there exists a method for constructing a Banach space which lies between

A0∩A1 and A0 +A1.

Definition 43.6.3 Let 1 ≤ q < ∞,0 < θ < 1. Define (A0,A1)θ ,q to be those elements ofA0 +A1,a, such that

||a||θ ,q ≡

[∫∞

0

(t−θ K (t,a,A0,A1)

)q dtt

]1/q

< ∞.

Theorem 43.6.4 (A0,A1)θ ,q is a normed linear space satisfying

A0∩A1 ⊆ (A0,A1)θ ,q ⊆ A0 +A1, (43.6.26)

with the inclusion maps continuous, and((A0,A1)θ ,q , ||·||θ ,q

)is a Banach space. (43.6.27)

If a ∈ A0∩A1, then

||a||θ ,q ≤

(1

qθ (1−θ)

)1/q

||a||θ1 ||a||1−θ

0 . (43.6.28)

If A0 ⊆ A1 with ||·||0 ≥ ||·||1, then

A0∩A1 = A0 ⊆ (A0,A1)θ ,q ⊆ A1 = A0 +A1.

Also, if bounded sets in A0 have compact closures in A1 then the same is true if A1 isreplaced with (A0,A1)θ ,q. Finally, if

T ∈L (A0,B0) ,T ∈L (A1,B1), (43.6.29)