24.9. AN EXAMPLE OF POLISH SPACE 687

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.

Now ∫ ∣∣∣∣r 1p′ h∣∣∣∣p′ dµ =

∫|h|p

′rdµ = ∥h∥p′

Lp′ (µ̃)< ∞.

Thus∥∥∥∥r

1p′ h∥∥∥∥

Lp′ (µ)= ∥h∥Lp′ (µ̃) =

∥∥∥Λ̃

∥∥∥ = ∥Λ∥ and represents Λ in the appropriate way. If

p = 1, then 1/p′ ≡ 0. Now consider the existence of r. Since the measure space is σ finite,there exist {Ωn} disjoint, each having positive measure and their union equals Ω. Thendefine

r (ω)≡∞

∑n=1

1n2 µ(Ωn)

−1XΩn (ω)

This proves the Lemma.

Theorem 24.8.6 (Riesz representation theorem) Let (Ω,S ,µ) be σ finite and letX ′ have the Radon Nikodym property. Then for Λ ∈ (Lp(Ω;X ,µ))′, p ≥ 1 there existsa unique h ∈ Lq(Ω,X ′,µ), L∞(Ω,X ′,µ) if p = 1 such that Λ f =

∫h( f )dµ. Also ∥h∥ =

∥Λ∥. (∥h∥= ∥h∥q if p > 1, ∥h∥∞ if p = 1). Here 1p +

1q = 1.

Proof: The above lemma gives the existence part of the conclusion of the theorem.Uniqueness is done as before.

Corollary 24.8.7 If X ′ is separable, then for (Ω,S ,µ) a σ finite measure space,

(Lp (Ω;X))′ ∼= Lp′ (Ω;X ′

).

Corollary 24.8.8 If X is separable and reflexive, then for (Ω,S ,µ) a σ finite measurespace,

(Lp (Ω;X))′ ∼= Lp′ (Ω;X ′

).

Corollary 24.8.9 If X is separable and reflexive and (Ω,S ,µ) a σ finite measurespace,then if p ∈ (1,∞) , then Lp (Ω;X) is reflexive.

Proof: This is just like the scalar valued case.

24.9 An Example of Polish SpaceHere is an interesting example. Obviously L∞ (0,T,H) is not separable with the normedtopology. However, bounded sets turn out to be metric spaces which are complete andseparable. This is the next lemma. Recall that a Polish space is a complete separablemetric space. In this example, H is a separable real Hilbert space or more generally aseparable real Banach space.