21.11. EXERCISES 705

Therefore, the {uni } are all bounded. It follows that after taking subsequences k times there

exists a subsequence{

unk

}such that unk is a Cauchy sequence in Lp ([a,b] ;W ) . You simply

get a subsequence such that unki is a Cauchy sequence in W for each i. Then denoting this

subsequence by n,

||un−um||Lp(a,b;W ) ≤ ||un−un||Lp(a,b;W )

+ ||un−um||Lp(a,b;W )+ ||um−um||Lp(a,b;W )

≤ η

4+ ||un−um||Lp(a,b;W )+

η

4< η

provided m,n are large enough, contradicting 21.10.51.

21.11 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

af (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)+∫ t

0f ′ (s)ds