704 CHAPTER 24. THE BOCHNER INTEGRAL

24.13 Exercises1. Show L1 (R) is not reflexive. Hint: L1 (R) is separable. What about L∞ (R)?

2. If f ∈ L1 (Rn;X) for X a Banach space, does the usual fundamental theorem of cal-culus work? That is, can you say limr→0

1m(B(x,r))

∫B(x,r) f (t)dm = f (x) a.e.?

3. Does the Vitali convergence theorem hold for Bochner integrable functions? If so,give a statement of the appropriate theorem and a proof.

4. Suppose g ∈ L1 ([a,b] ;X) where X is a Banach space. Then if∫ b

a g(t)φ (t)dt = 0 forall φ ∈C∞

c (a,b) , then g(t) = 0 a.e. Show that this is the case. Hint: It will likelydepend on the regularity properties of Lebesgue measure.

5. Suppose f ∈ L1 (a,b;X) and for all φ ∈ C∞c (a,b) ,

∫ ba f (t)φ

′ (t)dt = 0.Then thereexists a constant, a ∈ X such that f (t) = a a.e. Hint: Let

ψφ (x)≡∫ x

a[φ (t)−

(∫ b

aφ (y)dy

)φ 0 (t)]dt, φ 0 ∈C∞

c (a,b) ,∫ b

aφ 0 (x)dx = 1

Then explain why ψφ ∈C∞c (a,b), ψ ′

φ= φ −

(∫ ba φ (y)dy

)φ 0. Then use the assump-

tion on ψφ . Next use the above problem. Verify that f (y) =∫ b

a f (t)φ 0 (t)dt a.e. y

6. Let f ∈ L1 ([a,b] ,X) . Then we say that the weak derivative of f is in L1 ([a,b] ,X) ifthere is a function denoted as f ′ ∈ L1 ([a,b] ,X) such that for all φ ∈C∞

c (a,b) ,

−∫ b

af (t)φ

′ (t)dt =∫ b

af ′ (t)φ (t)dt

Show that this definition is well defined. Next, using the above problems, showthat if f , f ′ ∈ L1 ([a,b] ,X) , it follows that there is a continuous function, denoted byt→ f̂ (t) such that f̂ (t) = f (t) a.e. t and f̂ (t) = f̂ (a)+

∫ t0 f ′ (s)ds. Thus, unlike the

classical definition of the derivative, when a function and its derivative are both in L1,it has a representative f̂ which equals the function a.e. such that f̂ can be recoveredfrom its derivative. Recall the well known example of this not working out whichis based on the Cantor function of Problem 4 on Page 268. This function had zeroderivative a.e. and yet it climbed from 0 to 1 on the unit interval. Thus one could notrecover it from integrating its classical derivative. Incidentally, if the function hasa derivative everywhere, then you can recover it by taking the generalized Riemannintegral of the derivative, although the Lebesgue integral of this derivative might noteven be defined. This is in my book on single variable advanced calculus, but thisintegral is not discussed here.