280 CHAPTER 10. NORMED LINEAR SPACES

You should use the fundamental theorem of calculus from one variable calculus andthe definition of the norm to verify this. Recall that limt→sf (t) = l means that forall ε > 0, there exists δ > 0 such that if 0 < |t− s| < δ , then ∥f (t)− l∥

∞< ε . You

have to use the definition of a limit in order to establish that something is a limit.

3. 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 collection of func-tions 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 form

F (t)≡ y0 +∫ t

0f (s)ds

where f ∈ G . Show that F is uniformly equicontinuous. Hint: This is a reallyeasy problem if you do the right things. Here is the way you should proceed. Re-member the triangle inequality from one variable calculus which said that for a < b∣∣∣∫ b

a f (s)ds∣∣∣≤ ∫ b

a | f (s)|ds. Then∥∥∥∥∫ b

af (s)ds

∥∥∥∥∞

= maxi

∣∣∣∣∫ b

afi (s)ds

∣∣∣∣≤maxi

∫ b

a| fi (s)|ds≤

∫ b

a∥f (s)∥

∞ds

Reduce to the case just considered using the assumption that these f are bounded.

4. Let V be a vector space with basis {v1, · · · ,vn}. For v ∈ V, denote its coordinatevector as v = (α1, · · · ,αn) where v = ∑

nk=1 αkvk. Now define ∥v∥ ≡ ∥v∥

∞. Show

that this is a norm on V .

5. Let (X ,∥·∥) be a normed linear space. A set A is said to be convex if whenever x,y ∈A the line segment determined by these points given by tx+(1− t)y for t ∈ [0,1] isalso in A. Show that every open or closed ball is convex. Remember a closed ballis D(x,r)≡ {x̂ : ∥x̂−x∥ ≤ r} while the open ball is B(x,r)≡ {x̂ : ∥x̂−x∥< r}.This should work just as easily in any normed linear space.

6. A vector v is in the convex hull of a nonempty set S if there are finitely many vectorsof S,{v1, · · · ,vm} and nonnegative scalars {t1, · · · , tm} such that

v =m

∑k=1

tkvk,m

∑k=1

tk = 1.

Such a linear combination is called a convex combination. Suppose now that S ⊆V,a vector space of dimension n. Show that if v = ∑

mk=1 tkvk is a vector in the convex