682 CHAPTER 21. THE BOCHNER INTEGRAL

21.8 The Riesz Representation TheoremThe Riesz representation theorem for the spaces Lp (Ω;X) holds under certain conditions.The proof follows the proofs given earlier for scalar valued functions.

Definition 21.8.1 If X and Y are two Banach spaces, X is isometric to Y if there existsθ ∈L (X ,Y ) such that

∥θx∥Y = ∥x∥X .

This will be written as X ∼= Y . The map θ is called an isometry.

The next theorem says that Lp′ (Ω;X ′) is always isometric to a subspace of (Lp (Ω;X))′

for any Banach space, X .

Theorem 21.8.2 Let X be any Banach space and let (Ω,S ,µ) be a finite measure space.Let p ≥ 1 and let 1/p+ 1/p′ = 1.(If p = 1, p′ ≡ ∞.) Then Lp′ (Ω;X ′) is isometric to asubspace of (Lp (Ω;X))′. Also, for g ∈ Lp′ (Ω;X ′),

sup|| f ||p≤1

∣∣∣∣∫Ω

g(s)( f (s))dµ

∣∣∣∣= ∥g∥p′ .

Proof: First observe that for f ∈ Lp (Ω;X) and g ∈ Lp′ (Ω;X ′),

s→ g(s)( f (s))

is a function in L1 (Ω). (To obtain measurability, write f as a limit of simple functions.Holder’s inequality then yields the function is in L1 (Ω).) Define

θ : Lp′ (Ω;X ′

)→ (Lp (Ω;X))′

by

θg( f )≡∫

Ω

g(s)( f (s))dµ.

Holder’s inequality implies∥θg∥ ≤ ∥g∥p′ (21.8.34)

and it is also clear that θ is linear. Next it is required to show ∥θg∥= ∥g∥.This will first be verified for simple functions. Let

g(s) =m

∑i=1

c∗i XEi (s)

where c∗i ∈ X ′, the Ei are disjoint and ∪mi=1Ei = Ω. Then ∥g∥ ∈ Lp′ (Ω;R), ∥g(s)∥ =

∑mi=1 ∥c∗i ∥XEi (s) .

Let h(s)≡ ∥g(s)∥p′−1 /∥g∥p′−1p′ . Then

∫Ω

∥g(s)∥X ′ h(s)dµ =∫

Ω

∥g(s)∥p′

X ′

∥g∥p′−1p′

dµ = ∥g∥Lp′ (Ω;X ′) (21.8.35)