686 CHAPTER 21. THE BOCHNER INTEGRAL

there exists a unique h ∈ Lp′(Ω;X ′), L∞(Ω;X ′) if p = 1 such that

Λ f =∫

h( f )dµ.

Also ∥h∥= ∥Λ∥. (∥h∥= ∥h∥p′ if p > 1, ∥h∥∞ if p = 1). Here

1p+

1p′

= 1.

Proof: First suppose r exists as described. Also, to save on notation and to emphasizethe similarity with the scalar case, denote the norm in the various spaces by |·|. Define anew measure µ̃ , according to the rule

µ̃ (E)≡∫

Erdµ. (21.8.40)

Thus µ̃ is a finite measure on S . Now define a mapping, η : Lp(Ω;X ,µ)→ Lp(Ω;X , µ̃)

by η f = r−1p f . Then

∥η f∥pLp(µ̃)

=∫ ∣∣∣r− 1

p f∣∣∣p rdµ = ∥ f∥p

Lp(µ)

and so η is one to one and in fact preserves norms. I claim that also η is onto. To see this,let g ∈ Lp(Ω;X , µ̃) and consider the function, r

1p g. Then∫ ∣∣∣r 1

p g∣∣∣p dµ =

∫|g|p rdµ =

∫|g|p dµ̃ < ∞

Thus r1p g ∈ Lp (Ω;X ,µ) and η

(r

1p g)= g showing that η is onto as claimed. Thus η is

one to one, onto, and preserves norms. Consider the diagram below which is descriptive ofthe situation in which η∗ must be one to one and onto.

h,Lp′ (µ̃) Lp (µ̃)′ , Λ̃

η∗

→ Lp (µ)′ ,Λ

Lp (µ̃)η

← Lp (µ)

Then for Λ ∈ Lp (µ)′ , there exists a unique Λ̃ ∈ Lp (µ̃)′ such that η∗Λ̃ = Λ,∥∥∥Λ̃

∥∥∥ = ∥Λ∥ .By the Riesz representation theorem for finite measure spaces, there exists a unique h ∈Lp′ (µ̃)≡ Lp′ (Ω;X ′, µ̃) which represents Λ̃ in the manner described in the Riesz represen-tation theorem. Thus ∥h∥Lp′ (µ̃) =

∥∥∥Λ̃

∥∥∥= ∥Λ∥ and for all f ∈ Lp (µ) ,

Λ( f ) = η∗Λ̃( f )≡ Λ̃(η f ) =

∫h(η f )dµ̃ =

∫rh(

r−1p f)

dµ

=∫

r1p′ h f dµ.