150 CHAPTER 5. FUNCTIONS ON NORMED LINEAR SPACES
5.13 Exercises1. Consider the metric space C ([0,T ] ,Rn) with the norm ∥f∥ ≡ maxx∈[0,T ] ∥f (x)∥∞
.Explain why the maximum exists. Show this is a complete metric space. Hint: If youhave {fm} a Cauchy sequence in C ([0,T ] ,Rn) , then for each x, you have {fm (x)}a Cauchy sequence in Rn. Recall that this is a complete space. Thus there existsf (x) = limm→∞fm (x). You must show that f is continuous. This was in the sectionon the Ascoli Arzela theorem in more generality if you need an outline of how thisgoes. Write down the details for this case. Note how f is in bold face. This means itis a function which has values in Rn. f (t) = ( f1 (t) , f2 (t) , · · · , fn (t)).
2. For f ∈C ([0,T ] ,Rn) , you define the Riemann integral in the usual way using Rie-mann sums. Alternatively, you can define it as∫ t
0f (s)ds =
(∫ t
0f1 (s)ds,
∫ t
0f2 (s)ds, · · · ,
∫ t
0fn (s)ds
)Then show that the following limit exists in Rn for each t ∈ (0,T ) .
limh→0
∫ t+h0 f (s)ds−
∫ t0 f (s)ds
h= f (t) .
You should use the fundamental theorem of calculus from one variable calculus andthe definition of the norm to verify this. As a review, in case we don’t get to it intime, for f defined on an interval [0,T ] and s ∈ [0,T ] , limt→sf (t) = l means thatfor all ε > 0, there exists δ > 0 such that if 0 < |t− s|< δ , then ∥f (t)− l∥
∞< ε .
3. Suppose f :R→ R and f ≥ 0 on [−1,1] with f (−1) = f (1) = 0 and f (x)< 0 for allx /∈ [−1,1] . Can you use a modification of the proof of the Weierstrass approximationtheorem for functions on an interval presented earlier to show that for all ε > 0 thereexists a polynomial p, such that |p(x)− f (x)| < ε for x ∈ [−1,1] and p(x) ≤ 0 forall x /∈ [−1,1]?
4. A collection of functions F of C ([0,T ] ,Rn) is said to be uniformly equicontinu-ous if for every ε > 0 there exists δ > 0 such that if f ∈ F and |t− s| < δ , then∥f (t)−f (s)∥
∞< ε . Thus the functions are uniformly continuous all at once. The
single δ works for every pair t,s closer together than δ and for all functions f ∈F .As an easy case, suppose there exists K such that for all f ∈F , ∥f (t)−f (s)∥
∞≤
K |t− s| . Show that F is uniformly equicontinuous. Now suppose G is a collectionof functions of C ([0,T ] ,Rn) which is bounded. That is, ∥f∥= maxt∈[0,T ] ∥f (t)∥∞
<M < ∞ for all f ∈ G . Then let F denote the functions which are of the formF (t) ≡ y0 +
∫ t0 f (s)ds where f ∈ G . Show that F is uniformly equicontinuous.
Hint: This is a really easy problem if you do the right things. Here is the wayyou should proceed. Remember the triangle inequality from one variable calcu-lus which said that for a < b
∣∣∣∫ ba f (s)ds
∣∣∣ ≤ ∫ ba | f (s)|ds. Then
∥∥∥∫ ba f (s)ds
∥∥∥∞
=
maxi
∣∣∣∫ ba fi (s)ds
∣∣∣≤maxi∫ b
a | fi (s)|ds≤∫ b
a ∥f (s)∥∞ds. Reduce to the case just con-
sidered using the assumption that these f are bounded.
5. Suppose F is a set of functions in C ([0,T ] ,Rn) which is uniformly bounded anduniformly equicontinuous as described above. Show it must be totally bounded.