16.2. FUNCTIONS OF MANY VARIABLES 443

Therefore,

≤m

∑k=0

(mk

)xk (1− x)m−k ε

2+2∥ f∥

∞ ∑(k−mx)2≥m2δ

2

(mk

)xk (1− x)m−k

≤ ε

2+2∥ f∥

∞

1

m2δ2

m

∑k=0

(mk

)(k−mx)2 xk (1− x)m−k

≤ ε

2+2∥ f∥

∞

14

m1

δ2m2

< ε

provided m is large enough. Thus ∥pm− f∥∞< ε when m is large enough. ■

Note that we do not need to have X be complete in order for this to hold. It would havesufficed to have simply let X be a normed linear space.

Corollary 16.1.3 If f ∈C ([a,b] ;X) where X is a normed linear space, then there exists asequence of polynomials which converge uniformly to f on [a,b].

Proof: Let l : [0,1] → [a,b] be one to one, linear and onto. Then f ◦ l is contin-uous on [0,1] and so if ε > 0 is given, there exists a polynomial p such that for allx∈ [0,1] ,∥p(x)− f ◦ l (x)∥< ε . Therefore, letting y= l (x) , it follows that for all y∈ [a,b] ,∥∥p

(l−1 (y)

)− f (y)

∥∥< ε. ■

The exact form of the polynomial is as follows.

p(x) =m

∑k=0

(mk

)xk (1− x)m−k f

(l(

km

))

p(l−1 (y)

)=

m

∑k=0

(mk

)(l−1 (y)

)k (1− l−1 (y)

)m−kf(

l(

km

))(16.1)

Here is a corollary.

Corollary 16.1.4 Let f be a continuous function defined on [−M,M] with f (0) = 0. Thenthere exists a sequence of polynomials {pm}, pm (0) = 0 and

limm→∞∥pm− f∥

∞= 0

Proof: From Corollary 16.1.3 there exists a sequence of polynomials { p̂m} such that∥ p̂m− f∥

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

16.2 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)