132 CHAPTER 5. FUNCTIONS ON NORMED LINEAR SPACES

Corollary 5.6.3 If f ∈C ([a,b] ;X) where X is a normed linear space, then there existsa sequence of polynomials which converge uniformly to f on [a,b]. The mth term of thissequence is ∑

mk=0 qk (y) f

(l( k

m

))where l : [0,1]→ [a,b] be one to one, linear and onto and

q0 (a) = 1 and if k ̸= 0,qk (a) = 0 and qm (b) = 1 and if k ̸= m, then qk (b) = 0.

Proof: Let l : [0,1]→ [a,b] be one to one, linear and onto. Then f ◦ l is continuous on[0,1] and so if ε > 0 is given, if m large enough, then for all x ∈ [0,1] ,∥∥∥∥∥ m

∑k=0

q̂k (x) f(

l(

km

))− f ◦ l (x)

∥∥∥∥∥< ε

where q̂0 (0) = 1 and q̂k (0) = 0 for k ̸= 0, q̂m (1) = 1, q̂k (1) = 0 if k ̸= m. Therefore, for ally ∈ [a,b] , ∥∥∥∥∥ m

∑k=0

q̂k(l−1 (y)

)f(

l(

km

))− f (y)

∥∥∥∥∥< ε

Let qk (y)≡ q̂k(l−1 (y)

). ■

As another corollary, here is the version which will be used in Stone’s generalizationlater.

Corollary 5.6.4 Let f be a continuous function defined on [−M,M] with f (0) = 0.Then there is a sequence of polynomials {pm}, pm (0) = 0 and

limm→∞∥pm− f∥

∞= 0

Proof: From Corollary 5.6.3 there exists a sequence of polynomials {p̂m} such that∥p̂m− f∥

∞→ 0. Simply consider pm = p̂m− p̂m (0). ■

5.7 Functions of Many VariablesFirst note that if h : K×H→R is a real valued continuous function where K,H are compactsets in metric spaces,

maxx∈K

h(x,y)≥ h(x,y) , so maxy∈H

maxx∈K

h(x,y)≥ h(x,y)

which implies maxy∈H maxx∈K h(x,y)≥max(x,y)∈K×H h(x,y) . The other inequality is alsoobtained.

Let f ∈C (Rp;X) where Rp = [0,1]p . Then let x̂p ≡ (x1, ...,xp−1) . By Theorem 5.6.2,if n is large enough,

maxxp∈[0,1]

∥∥∥∥∥ n

∑k=0f

(·, k

n

)(nk

)xk

p (1− xp)n−k−f (·,xp)

∥∥∥∥∥C([0,1]p−1;X)

<ε

2

Now f(·, k

n

)∈C (Rp−1;X) and so by induction, there is a polynomial pk (x̂p) such that

maxx̂p∈Rp−1

∥∥∥∥pk (x̂p)−(

nk

)f

(x̂p,

kn

)∥∥∥∥X<

ε

(n+1)2