Kenneth Kuttler

21.8. THE RIESZ REPRESENTATION THEOREM 683

Also h ∈ Lp (Ω;R) and

∫Ω

|h(s)|p dµ =∫

Ω

∥g(s)∥p′

∥g∥p′p′

=∥g∥p′

∥g∥p′= 1

so ∥h∥p = 1. Since the measure space is finite, h ∈ L1 (Ω;R).Now let di be chosen such that

c∗i (di)≥ ∥c∗i ∥X ′ − ε/∥h∥L1(Ω)

and ∥di∥X = 1. Let

f (s)≡m

∑i=1

dih(s)XEi (s) .

Thus f ∈ Lp (Ω;X) and ∥ f∥Lp(Ω;X) = 1. This follows from

∥ f∥pp =

∫Ω

∑i=1∥di∥p

X |h(s)|p XEi (s)dµ =

∑i=1

(∫Ei

|h(s)|p dµ

)= 1

Also

∥θg∥ ≥ |θg( f )|=∣∣∣∣∫

Ω

g(s)( f (s))dµ

∣∣∣∣≥∣∣∣∣∣∫

Ω

∑i=1

(∥c∗i ∥X ′ − ε/∥h∥L1(Ω)

)h(s)XEi (s)dµ

∣∣∣∣∣Then from 21.8.35

≥∣∣∣∣∫

Ω

∥g(s)∥X ′ h(s)dµ

∣∣∣∣− ε

∣∣∣∣∫Ω

h(s)/∥h∥L1(Ω) dµ

∣∣∣∣= ∥g∥Lp′ (Ω;X ′)− ε.

Since ε was arbitrary, ∥θg∥ ≥ ∥g∥ and from 21.8.34 this shows equality holds whenever gis a simple function.

In general, let g ∈ Lp′ (Ω;X ′) and let gn be a sequence of simple functions convergingto g in Lp′ (Ω;X ′). Such a sequence exists by Lemma 21.1.2. Let gn (s)→ g(s) ,∥gn (s)∥ ≤2∥g(s)∥ . Then each gn is in Lp′ (Ω;X ′) and by the dominated convergence theorem theyconverge to g in Lp′ (Ω;X ′). Then for ∥·∥ the norm in (Lp (Ω;X))′ ,

∥θg∥= limn→∞∥θgn∥= lim

n→∞∥gn∥= ∥g∥.

This proves the theorem and shows θ is the desired isometry.

Theorem 21.8.3 If X is a Banach space and X ′ has the Radon Nikodym property, then if(Ω,S ,µ) is a finite measure space,

(Lp (Ω;X))′ ∼= Lp′ (Ω;X ′

)and in fact the mapping θ of Theorem 21.8.2 is onto.

21.8. THE RIESZ REPRESENTATION THEOREM 683Also h € L? (Q;R) and[wcran= [ le(s)l” _ Mglly _P= Melle _lel Isllyso ||h||,, = 1. Since the measure space is finite, h € L!(Q;R).Now let d; be chosen such thatcj (di) 2 lei IIx — €/ IlAllzi(Q)and ||dj||y = 1. Letf(s)= yan (s) 2x, (s).Thus f € L? (Q;X) and ||f||;»(9:x) = 1. This follows fromrp = JY blo P Hi (o)au = ( f im(siPae) =Alsolos > lee A= [ e0r(r(o) au >J (lief —€/|lhl,¢@)) h(s) Zz, (s)duThen from 21.8.35> | [ letilem(syan|-e[¥6)/lPtacaya| = Islan) -©Since € was arbitrary, ||@g|| > ||g|| and from 21.8.34 this shows equality holds whenever gis a simple function.In general, let g € LY (Q;X") and let g, be a sequence of simple functions convergingto gin L” (Q;X’). Such a sequence exists by Lemma 21.1.2. Let gn (s) > g(s),|Ign(s)|| <2||g(s)||. Then each gy is in L”’ (Q;X’) and by the dominated convergence theorem theyconverge to g in L”’ (Q:X’). Then for ||-|| the norm in (L? (Q:X))',| 6 || = Lim ||@gn|| = lim |Ign|| = [eIThis proves the theorem and shows @ is the desired isometry. ffTheorem 21.8.3 If X is a Banach space and X' has the Radon Nikodym property, then if(Q,.%, LL) is a finite measure space,(L? (Q;X))' = L” (Q;X’)and in fact the mapping 9 of Theorem 21.8.2 is onto.