A GENERALIZATION OF THE WEIERSTRASS

APPROXIMATION THEOREM

APPROVED:

/! Major Professor

Fro re nor Frofess

, F^JL lrectQaf) of the Depart!

Directcjj?} of the Department.of Mathematics

Dean qf the Graduate School

Murchison, Jo Denton, A generalization of the Weierstrass

Approximation Theorem. Master of Science (Mathematics),

August, 1972, 36 pp., bibliography, H titles.

A presentation of the Weierstrass approximation theorem

and the Stone-Weierstrass theorem and a comparison of these

two theorems are the objects of this thesis. George F.

Simmons' Introduction to Topology and Modern Analysis is the

source used for the Weierstrass approximation theorem, and

Leopoldo Nachbin's Elements of Approximation Theory is the

primary source used for the Stone-Weierstrass theorem.

The material is presented in four chapters: Intro-

duction, The Weierstrass Approximation Theorem, The Kakutani-

Stone Theorem, and The Stone-Weierstrass Theorem. Definitions

and basic theorems which are assumed are presented in Chap-

ter I.

Chapter II contains a proof of the Weierstrass approxi-

mation theorem. The method of proof selected for this theorem

makes use of the Bernstein polynomials.

The definitions and theorems in Chapter III are necessary

to the proof of the Kakutani-Stone theorem on the closure of a

lattice. The Kakutani-Stone theorem is essential to the proof

of the Stone-Weierstrass theorem. Chapter III begins with a

discussion of topological vector spaces. A topology T , may be

defined on a vector space V by a family of seminorms T which

determines a subbasis for the topology. It is shown that V is

a topological vector space with respect to Tp. It is also

demonstrated that if a topology Tp is defined on an algebra

A by a family of algebra seminorms r, then A is a topological

algebra with respect to T . Next a family of algebra semi-

norms T is defined on C(E; K), the continuous K-valued func-

tions defined on a completely regular space E, where K is

used to denote the real number system R or the complex number

system C. It is shown that the algebra C(E; K) with the topol-

ogy Tp determined by F is a topological algebra. Once these

results are developed, it is possible to prove the Kakutani-

Stone theorem on the closure of a lattice L, where L C C(E; R)

and T is the topology defined on C(E; R).

Chapter IV presents the Stone-Weierstrass theorem on the

closure of an algebra and contains lemmas necessary to the

proof of this theorem. This'chapter also contains a discus-

sion of the Stone-Weierstrass theorem as a generalization of

the Weierstrass approximation theorem. There are two cases

to be considered when proving the Stone-Weierstrass theorem.

The first case considers the closure of a subalgebra A of

C(E; R). The Kakutani-Stone theorem is used in the proof

of the first case. Next the closure of a subalgebra B of

C(E; C) is considered. The thesis concludes with a corollary

to the Stone-Weierstrass theorem.

A GENERALIZATION OF THE WEIERSTRASS

APPROXIMATION THEOREM

THESIS

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of the Requirements

For the Degree of

MASTER OF SCIENCE

By

Jo Denton Murchison, B. A.

Denton, Texas

August, 1972

TABLE OF CONTENTS

Chapter Page

I. INTRODUCTION . . . . . 1

II. THE WEIERSTRASS APPROXIMATION THEOREM . . . . 11

III. THE KAKUTANI-STONE THEOREM 15

IV. THE STONE-WEIERSTRASS THEOREM 26

BIBLIOGRAPHY 36

i n

CHAPTER I

INTRODUCTION

A presentation of the Weierstrass approximation theorem

and the Stone-Weierstrass theorem and a comparison of these

two theorems are the objects of this thesis. Definitions and

basic theorems which will be assumed are stated in Chapter I.

Chapter II consists of a proof of the Weierstrass approxima-

tion theorem. The purpose of Chapter III is to present the

Kakutani-Stone theorem, which is required in the proof of the

Stone-Weierstrass theorem. The many definitions and theorems

stated in Chapter III are necessary for proving the Kakutani-

Stone theorem. In Chapter IV, the Stone-Weierstrass theorem

is presented and is discussed as a generalization of the

Weierstrass approximation theorem.

The real number system will be denoted by R, and the

complex number system will be signified by C. In order to

indicate either R or C without being specific, K will be

used.

Definition 1.1; A nonempty set X of elements is called

a vector space over K if there is an operation

(x, y) e X x X + x + y e X and an operation

(a, x ) e K x X - * - a x e X , which satisfy the following condi-

tions :

(1) x + y = y + x;

(2) (x + y) + z = x + (y + z) ;

(3) there exists a unique element 0 in X such that

x + 0 = x for every x e X;

(4) to each element x in X, there corresponds a unique

element (-x) in X such that x + (-x) = 0;

(5) a(x + y) = ax + ay, where a e K-, x, y e X;

(6) (a + 3) x = ax + 8x, where a, $ e K; x e X;

(7) a(8x) = (a3)x, where a, 3 e K; x e X; and

(8) l*x = x.

Definition 1.2: A subspace S of the vector space X over

K is a nonempty subset of X such that:

(1) if a and b are in S, then a + b e S; and

(2) if a e S and a e K, then aa e S.

Definition 1.3: Let V be a vector space over R. A

vector of the form aJXi + . . . + arXr, where X^ e V and

aj, . . . , a are arbitrary real numbers, is called a linear

combination of the vectors X1, . . . , Xr.

Definition 1.4: Let V be a vector space over R. If it

is possible to find r vectors Xj, . . . , X of V such that

every vector X of V can be written as a linear combination

X = a X + . . . + arXr, then the vectors Xj, . . . , Xr are

called a set of generators of V. The notation

V = L{Xi, . . . , X ,} is used to denote that the vector space

V is generated by the vectors X , . . . , Xp.

Definition 1.5: The vectors Xj , . . . , Xr of a vector

space V over R are said to be linearly dependent if there

exist real numbers a2 , . . . s ar»

no"t a H zero, such that

ajXx + . . . + arXr =0. If the vectors are not linearly

dependent, they are said to be linearly independent.

Definition 1.6: A set of vectors Xl, . . . , Xk of a

vector space V over R is called a basis for V if

(1) V = L{Xi, . . . , Xk}; and

(2) Xi, . . . , X^ are linearly independent.

Theorem 1.7: A set of linearly independent vectors

Xj, . . . , X^ of a vector space V over R is a basis for V

if and only if the maximum number of linearly independent

vectors of V is k.

Definition 1.8: The dimension of a vector space V over

R is the number of vectors in a basis for V.

Theorem 1.9: Let (a, b), (c, d) e R2 and a e R. Define

(a, b) + Cc, d) = (a+ c, b + d) and a(a, b) = (aa, ab).

(1) Then R2 with respect to the operations stated above

is a vector space over R of dimension two.

(2) If A is a subspace of R2 where A i R2 and A i 0,

then A has dimension one.

(3) Any subspace A of R2 having dimension one is a line

in R2 containing the point (0, 0).

Definition 1.10: A vector space A over K is called

an algebra if there is an operation (x, y) e A x A xy e A

which satisfies the following conditions:

(1) x(yz) = (xy)z;

(2) x(y + z) = xy + xz, and (x + y)z = xz + yz;

(3) a(xy) = (ax)y = x(ay) for every a e K.

Definition 1.11: A subset A of an algebra B is a sub-

algebra of B if and only if A is an algebra with respect to

the same operations defined on B.

Definition 1.12: If A and B are algebras which are vec-

tor spaces over K, then the mapping f: A + B is called an

algebra homomorphism if and only if

(1) f(x + y) = f(x) + f(y) for all x, y e A;

(2) f(cx) = cf(x) for all x e A and all c e K; and

(3) f(xy) = f(x)f(y) for all x, y e A.

Theorem 1.13: Let A and B be algebras which are vector

spaces over K. If the mapping f: A + B is an algebra homo-

morphism and C is a subalgebra of A, then f(C) is a sub-

algebra of B.

Definition 1.14: A topological space (X, T) is a set

X and a family T of subsets of X satisfying the following

axioms:

(1) 4> e T and X e T;

(2) if A and B are in T, then A 0 B e T; and

(3) if A^ e T for every <5 e I, then U{Ag| 6 e 1} e T.

The members of T are called open sets, and T is called a

topology on X.

Theorem 1.15: Let (X, T) be a topological space, Y C X,

and Ty = {U A Y| U e T}. Then (Y, T^) is a topological space,

Definition 1.16: The space (Y, T^) is called a subspace

of (X, T), and the topology Ty is called the relative topology

on Y.

Definition 1.17: A family B of subsets of a set X is a

base for the open sets of a topology T on X if it has the

following properties:

(1) for each x e X, there is a Bx e B such that x e Bx;and

(2) if B1} B2 e B and if x e Bj f| B2 , then there is a

B3 e B such that x e B3 C Bj H B2.

The family of open sets of T is defined as follows: A nonvoid

set 0 is open if and only if it is the union of some subcol-

lection of B. The topology T is called the topology generated

by the base B.

Definition 1.18: A family S of subsets of a set X is a

subbase for a topology T on X if and only if for each x e X

there is Sx e S such that x e Sx. The topology T is that

topology generated by the base B consisting of all finite

intersections of sets in S.

Theorem 1.19: A necessary and sufficient condition for

a family B of subsets of X to be a base for a given topology

T on X is the following: B C T, and given any 0 e T and

x e 0, there is a B e B such that x e B C 0. X X

Definition 1.20: A collection B of open sets of a point

x in a topological space (X, T) is said to be a basis at x

if and only if for each open set 0 containing x there exists

0X e B such that x e 0X C 0.

Definition 1.21: A collection S of open sets of a point

x in a topological space (X, T) is said to be a subbasis of

open neighborhoods at x if and only if the collection of all

finite intersections of members of S is a basis at x.

Theorem 1.22: If (X, T) and (Y, S) are two topological

spaces, a topology on the product X x Y is defined by taking

as a base the collection of all sets of the form OjX 02?

where Oj e T and 02 e S. This is called the product topology

for X x Y. Let A = {X x G2| G2 e S} and B = {G1 x Y| Gx e T}.

Then A U B is a subbase for the product topology.

In this thesis, the understood topology for X x Y is the

product topology.

Definition 1.23: A collection U of open sets in a topo-

logical space is an open covering for a set K if K is con-

tained in the union of the sets in U.

Definition 1.2M-: A topological space (X, T) is said to

be compact if every open covering U of X has a finite sub-

covering, that is, if there is a finite collection

{0, , . . ; , CL} C U such that X = .U,0•. A subset K of a n i = i x

topological space is called compact if it is compact as a

subspace of X.

Theorem 1.25: A subset K of X is compact if and only

if every covering U of K by open sets of X has a finite sub-

covering.

Theorem 1.26: The union of two compact subsets of a

topological space is compact.

Definition 1.27: A subset A of a topological space (X, T)

is called closed if and only if its complement X - A is open.

Definition 1.28: The closure A of a set A is the inter-

section of all closed sets containing A.

Theorem 1.29: If (X, T) and (Y, S) are topological

spaces, A C X, and B C. Y, then A x B = A x B.

Definition 1.30: A point x is a limit point of a set A

if and only if every open set containing x contains a point

of A different from x. The set of all limit points of a set

A is called the derived set of A and is denoted by A'.

Theorem 1.31: Let (X, T) be a topological space and

A C X. Then A = A U A'.

Theorem 1.32: If each member of a basis B at x contains a x

point of a set A different from x, then x is a limit point of A.

Definition 1.33: Let (X, T) be a topological space and

A C X . Then A is said to be dense in X if X = X.

Definition 1.3H: A mapping f of a topological space

(X, T) into a topological space (Y, S) is said to be contin-

uous if the inverse image of every open set is open.

Definition 1.35: A mapping f of a topological space (X, T)

into a topological space (Y, S) is said to be continuous at

the point x e X if to every open neighborhood V of f(x) there

corresponds an open neighborhood W of x such that f(W) C. V.

Theorem 1.36: A mapping f of a topological space (X, T)

into a topological space (Y, S) is continuous if and only if

f is continuous at every point of X.

Theorem 1.37: Let (X, T) and (Y, S) be topological

spaces, f: X -»• Y a mapping, a e X, b = f(a), and W a sub-

basis of open neighborhoods at b. If for every V e W,

there is an open neighborhood G of a such that f(G) C. V,

then f is continuous at a.

Theorem 1.38: Let (X, T) and (Y, S) be topological

spaces. If A C X and f: X Y is continuous, then

(f|A): A -*• Y is continuous, where f|A is the restriction of

f to A.

Theorem 1.39: Let f be a continuous real-valued func-

tion on a compact space X. Then f is bounded and assumes

its maximum and minimum.

Theorem 1.40: If f is a continuous K-valued function

defined on a topological space (X, T), then the function g,

where g(x) = |f(x)|, is a continuous real-valued function

defined on (X, T).

Definition 1.41: A metric space (X,d) is a set X

together with a real-valued function d defined on X x X

which satisfies the following axioms:

(1) d(x, y) >_ 0 for all x, y e X, and d(x, x) = 0;

(2) d(x, y) = 0 implies x = y;

(3) d(x, y) = d(y, x) for every x, y e X; and

(4) d(x, z) £ d(x, y) + d(y, z) for all x, y, z e X.

Definition 1.42: A function f from a metric space

(X, d) to a metric space (Y, d1) is said to be continuous

at x if for every £ > 0, there is 6 > 0, such that if

d(x, y) < 6, then d'(f(x), f(y)) < £. The function f is

called continuous if it is continuous at each x e X.

Definition 1.43: Let f be a mapping from the metric

space (X, d) to the metric space (Y, d'). Then f is uni-

formly continuous if given £ > 0, there exists 6 > 0 such

that if X p x2 e X and d(xx, x2) < 6, then d'CfCXj), f(x2)) < 5.

Theorem 1.44; If a real-valued function f is defined

and continuous on a closed and bounded set F of real num-

bers, it is uniformly continuous on F.

Definition 1.45; A one-to-one mapping f of a topolo-

gical space (X, T) onto a topological space (Y, S) is called

a homeomorphism between (X, T) and (Y, S) if f and f"1 are

continuous.

Theorem 1.46: If (X, T) and (Y, S) are topological

spaces and f: X + Y is a one-to-one, onto mapping, the fol-

lowing are equivalent:

(1) f is a homeomorphism.

(2) If G C X, then f(G) is open in Y if and only if

G is open in X.

(3) If FCZ X, then f(F) is closed in Y if and only if

F is closed in X.

Theorem 1.47: Let C(E; K) denote the continuous K-

valued functions defined on a topological space (E, T).

Then C(E; K) is a vector space over K and is also an algebra.

10

Theorem 1.U8; Let f, g e C(E; R). Then

sup(f, g) = (f + g + jf - g[)/2 and

inf (f, g) = (f + g - |f - g|) / 2.

Theorem 1.U9: If f e C(E; C), then f = g + ih,

where g, h e C(E; R).

Definition 1.50: If f = g + ih e C(E; C), then f = g - ih,

where f is the complex-conjugate of f.

Definition 1.51: If f is a continuous complex function *P + iF"

defined on a topological space (X, T), then Rf = — - — is

called the real part of f.

Definition 1.52: A completely regular space E is a

Hausdorff space such that, for any a e E and for any closed

subset F C E not containing a, there is a continuous real-

valued function f on E, such that 0 < f < 1, f(a) = 1, and

fCx) = 0 for any x e F.

CHAPTER II

THE WEIERSTRASS APPROXIMATION THEOREM

Theorem 2.1 (Weierstrass Approximation Theorem): Let

f be a continuous real-valued function on a closed interval

[a, b], and let £ > 0. Then there is a polynomial p such

that |p(x) - f(x)| < § for all x in [a, b].

Proof: If a = b, then let p be the constant polynomial

defined by pCx) = f(a). In this case the theorem is proved.

Assume that a < b. Next it will be shown that it is suffi-

cient to prove the theorem for the case when a = 0 and

b = 1. Assume the theorem is proved for the case when

a = 0 and b = 1. The mapping x = [b - a]x' + a from [0, 1]

onto [a, b] is continuous. Thus the composition function

g(x') = f([b - a]x' + a) is a continuous real-valued func-

tion defined on [0, 1]. Therefore there exists a polynomial

p' defined on [0, 1] such that |g(x') - p'(x')| < £ for all

xT in [0, 1]. If this inequality is expressed in terms of

x, then |f(x) - p'([x - a3/[b - a3)| < 5 for all x in [a, b3.

Define a polynomial p by p(x) = p'([x - a3/Cb - a3).

Then|f(x) - pCx)| < £ for all x in [a, b]. Therefore, if the

theorem is true for the case when a = 0 and b = 1, then it is

true for the general case. Thus assume a = 0 and b = 1.

If n is a positive integer and k an integer such that

0 £ k •< n, then the binomial coefficient (£) is defined by

11

12

(£) = n!/k!(n --k)!. The polynomial Bn defined by

n n

B (x) = Z (v)x^(l - x)n~^f(k/n) is called a Bernstein poly-

n k=0 K

nomial associated with f. The theorem will be proved by

finding a Bernstein polynomial with the required property.

Let x e [0, 1]. By the binomial theorem,

n Z (Ox^Cl - x)n"k = [x + (1 - x)]n = 1. If this equality

k=0

is differentiated with respect to x, then

Z (R)Ckxk~1(l - x)n-k - (n - k)xk(l - x)11"3^1] = k=0

Z (£)xk "*"(1 - x)n k"~ (k - nx) = 0. Multiplying through by k=0

x(l - x) gives Z (g)xk(l - x)n~k(k - nx) =0. If this k=0

equality is differentiated with respect to x, then

Z (jc)C-nxk(l - x)n-k + xk_1(l - x)n"k~1(k - nx)2] = 0. k=0

Using the equality just stated and the fact that

Z C5c)xkC1 - x)n-k = 1, then k=0

Z (k)xk-:L(l - x)n-k~"''(k - nx)2 = n. Multiplying through k=0

by x(l - x) gives Z (Jc)xk(l - x)n~''c(k - nx)2 = nx(l - x). k=Q

Dividing both sides of this last equality by n2 gives

Z (5c)xk(l - x)n-k(x - k/n)2 = . Since k=0 n

13

n v Z (£)x (1 - x)n~k = 1, then f(x) - Bn(x) =

k=0

f(x)[ Z (jj)xk(l - x)n"k] - Z (£)xk(l - x)n-kf(k/n) = k=0 k=0

Z (£)xk(l - x)n"k[f(x) - f(k/n)]. Thus k=0

n i f(x) - B (x)1 < Z (. )xk(l - x)n |f(x) - f(k/n)|. Since

k=0

f is continuous on [0, 1], f is uniformly continuous on [0, 1].

Let £ > 0. Then there exists 6 > 0 such that if

|x - k/n| < 6, then |f(x) - f(k/n)| < £/2. Split

n , Z (R)x (1 - x)n |f(x) - f(k/n)| into two parts called

k=0 Z and Z'. Define Z as the sum of those terms for which

|x - k/n| < 6, and define Z' as the sum of the remaining

n «, terms. Since Z ( )xk(l - x)n = 1 and since |x - k/n| < 6

k=0 K

implies |f(x) - f(k/n)| < C/2, then Z < £/2. It will be

shown that if n is taken sufficiently large, then Z' can be

made less than £/2 independently of x. By Theorem 1.39, f

is bounded on [0, 1], Thus there exists a positive number

M such that |f(x)| <_ M for all x in [0, 1], Therefore

Z' < 2MZ(£)xk(l - x)n"k, where

Z(£)xk(l - x)n~k = Z" is taken over all k such that

|x - k/n | >_ 6. It will be shown that if n is taken suffi-

ciently large, then Z" can be made less than £/^M indepen-

dently of x. Since Z (Jc)xk(l - x)n~k(x - k/n)2 = k=0 n

14

and |x - k/n| >_• 6, then 62Z" < ~ x \ Thus — n

X ( 1 *" x) - • Since the maximal value of x(l - x) on

6 n

[0, 1] is 1/4, then Z" < for all x in [0, 1]. Let n

be any integer greater than M/52?. Then I" < £/4M,

Z* < 5/2, and |f(x) - Bn(x)| < Z + Z1 < ^ for all x in [0, 1].

Therefore the theorem is proved.

CHAPTER III

THE KAKUTANI-STONE THEOREM

Definition 3.1: A topological vector space E is a

vector space which at the same time is a topological space

such that the vector space operations

(x, y) e E x E - * x + y e E and (X, x) e K * E - * X x e E are

cont inuous .

Definition 3.2: A seminorm p on a vector space E is a

real-valued function on E such that: (1) p(x) >_ 0 for

x e E; (2) p(ax) = ]a|*p(x) for a e K, x e E; and

(3) p(x + y) <_ p(x) + pCy) for x, y e E.

Definition 3.3: Let p be a seminorm on the vector

space E. If a e E, r > 0, the open ball B„ _.(a) with center — x

a and radius r is the set of all x e E such that

p(x - a) < r.

Definition 3. U: Let T = be a. family of semi-

norms on the vector space E. The natural topology Tp de-

fined by T on E is introduced as follows. If a e E, E, > 0,

and i e I, then the collection of all subsets of E of the

form Bg is a subbasis for the topology on E.

Lemma 3.5: If p is a seminorm on the vector space E

and a e B^pCb), where a, b e E, E, > 0, then

C where 6 = £ - p(a - b) > 0. 0 »P bsP

15

16

Proof: Let x e Bg p(a). Then p(x - a) < 6, and

p(x - b) <_ p(x - a) + p(a - b) < 6 + p(a - b) = ?. Thus

x e Br (b).

Theorem 3.6: Let T = {pi>iei be a family of seminorms

on the vector space E. If a e E, £ > 0, and ix, . . . , in e I,

let V5, ij, . . • , in^a^ = B?,p;j_^

a^ H • • • H Bg,pin^a^

{x e E| pi (x - a) < • • • » Pin(x " a ) < ^ ' T h e n f o r

each fixed a e E the collection of all such

Vr 4 ,• (a) is a basis of open neighborhoods at a ip • • • * 1n

r Proof: Let a e E and 0 be an open set containing a.

for Tj,.

Then there exists a basic element A of Tp such that a e A =

Bf (bi) n . . . ft Be T3. (b ) c 0, where Si,Pi 11 ^n»Pin n

b , . . • , bn e E, > 0, . . . , gn > 0, and

ix, . . . , in c I. Let <5 = inf{6h = ?h - Pij/a ~ V I

h = l , . . . , n } > 0 . By Lemma 3.5, Bg . (a) Q. n'F1h

B^^p^Cb^), where h = 1, . . . , n. Since

B«.Pih(a) c B«h-Pih

(a)<: Beh.Pih(bh>' w h e r e h = ' ' ' ' n'

then a e i . (a) £ A £0. Therefore the collec-> i, . . . , in

tion of all such elements Vr -• (a) is a basis at a c,, x i, . . . xn

for Tr.

Corollary 3.7: For each fixed a e E, the collection of

all Br _ (a), where £ > 0, i e I, is a subbasis of open neigh-s' >Pi

borhoods at a.

17

Proof: The proof of this corollary follows directly

from Theorem 3.6 and Definition 1.21.

Theorem 3.8: If r = {pj^igj is a family of seminorms

on the vector space E, then E is a topological vector space

with respect to T ,.

Proof: It is necessary to prove continuity of

(x, y) e E x E -»• x + y e E at (a, b), where a, b e E. Let

c = a + b. By Corollary 3.7, the collection of all p(c),

where £ > 0, i e I, is a subbasis of open neighborhoods at c.

Consider a subbasic neighborhood B^ p.Cc) of c, where £ > 0,

i e I. By Theorem 1.22, B^/2,p.^a^ x B£/2,p.^ i s a n °Pen

subset of E x E containing (a, b). If x e B ( a ) . 5/2, Pi

y e b£/2, Pi^^' t h e n Pi^x + y - c) < p.j_(x - a) + Pj_(y - b) <

£/2 + 5/2 = £. Thus x + y e B^p^Cc), and by Theorem 1.37

the desired continuity is proved.

Next it is necessary to prove the continuity of

(X, x) e K x E Xx e E at (a, a), where a e K and a e E.

Set b = aa. Consider a subbasic neighborhood Br n.(b) of ' * i

b, where 5 > 0, i e I. Let y > 0, 6 > 0, and x e Bfijp.(a).

Let A = {0j|$-a| < y}. Then A x B^^p^(a) is open in

K x E. Let X e A. Then

p^CXx - aa) = p^CXCx - a) + (X - a)a] <_

|x|p^(x - a) + |X - a|p^(a). Since

|X| <_ | X — ot | + |a| < y + | a | , then

p^CXx - aa) < (y + |a|)6 + yp^(a) <_ £, provided y and <5 are

18

chosen sufficiently small. Thus Xx e Br _ (b). By Theorem

1.37, the desired continuity is proved. By Definition 3.1,

E is a topological vector space with respect to Tp.

Definition 3.9: The family T = ' fPi^gj °f seminorms on

the vector space E is directed if for any ix, i e I, there

are i e I, X e R, and X > 0 such that p ^ £ Xp^, p£2 £

Theorem 3.10: Let T = be a directed family of

seminorms on the vector space E. Then for each fixed a £ E

the collection of all Br ^.(a) is a basis of open neighbor-i

hoods at a with respect to T , where £ > 0, i e I.

Proof; Let a e E and 0 be an open set containing a.

By Definition 1.2 0 and Theorem 3.6, there exists a basic

element Vr A . _• (a) such that a e V,. • . (a) C,5-L15 # • • • 5

CO, where 5 > 0 and ix, . . . , in e I. Since is a

directed family of seminorms on E, for i , . . . , in e I

there are i e I, X e R, and X > 0 such that

Pi* < Xp• , . . . , p- < Xp-. Let 6 = £/X and x e Br (a). 11 •— ± 'Pi

Then p^(x - a) < 6 = ?/X, and Xp^Cx - a) < £. Since

p.- (x - a) < Xp.(x - a) < £, where 1 < t < n, then X-J- ' """"

x e Br _. (a), where 1 < t < n. Thus x e V- . . . », sPx-t — — s»ii» . • . ,inva;

and a e B* ^.(a) C Vr . • (a) C 0. By Definition ° >Pi ^ i > • • • » 1n

1.20, the collection of all B- (a), where £ > 0 and i e I, ^ 9 P

is a basis of open neighborhoods at a with respect to Tp.

19

Definition '3.11: A topological algebra A is an algebra

which at the same time is a topological space such that the

algebra operations (x, y) e A x A - * x + y e A ,

(X, x ) e K x A - * - X x e A , and (x, y) e A x A + x y e A are

continuous.

Definition 3.12: An algebra seminorm p on an algebra A

is a seminorm on A such that p(x*y) <_p(x)p(y) for x, y e A

and such that p(e) is either 1 or 0 in case A has a unit e.

Theorem 3.13: If T = {pi)iei is a family of algebra

seminorms on the algebra A, then A is a topological algebra

with respect to Tp.

Proof: By Theorem 3.8, A is a topological vector space

with respect to Tp. Thus (x, y) eAxA-»-x + y e A and

(X, x) e K x A + X x e A are continuous. Using Theorem 1.37,

it is possible to prove continuity of (x, y) e A x A -> xy e A

at (a, b), where a, b e A. Let c = ab. Since the collection

of all Br _ (c), where £ > 0, i e I, is a subbasis of open

neighborhoods at c, consider a subbasic neighborhood

Br ^ (c) of c, where E, > 0, i e I. Let { > 0, x c B. (a), ,P£ o >P£

and y e B„ (b). By Theorem 1.22, Br ^.(a) x B* ^.(b) is J 6,Pi 6>Pi °»Pi

open. Now p^Cxy - ab) = p^CxCy - b) + (x - a)b] <_

p^[x(y - b)] + piE(x - a)b] <_ p£(x)p. (y - b) + p^(x - a)p^(b).

Since p^(x) <_ P^(x - a) + p^(a) < 6 + p^(a), then

p^Cxy - c) < [6 + p^(a)]6 + 6p^(b) <_ £, provided that <5 is

chosen sufficiently small. Thus xy e Bf (c). Then the ^ » Pi ..

20

desired continuity is proved. Therefore A is a topological

algebra with respect to Tp.

Definition 3.14: If f is a continuous K-valued function

defined on a topological space X and if E is a nonempty com-

pact subset of X, let H^Hg = sup{|f(x)| |x e E}.

Theorem 3.15: Let E be a completely regular space.

Every nonempty compact subset F C E determines an algebra

seminorm Pp on C(E; K) , where Pp(f) = HfJIp and f e C(E; K).

Proof: First it is necessary to show that Pp is a real-

valued function and Pp(f) 0 for f t C(E; K). Suppose

f = g, where f, g e C(E; K). Then

1! f I[ P = sup { I f (x) | | x e F}, and ||g||p = sup{[g(x)| |x e F} .

Since f and g are continuous K-valued functions defined on

E, by Theorem 1.40 |f| and jg| are continuous real-valued

functions defined on E. By Theorem 1.38, (fI |F and |g| |F

are continuous. Thus by Theorem 1.39, the functions |f| |F

and |g| |F are bounded on F and assume their maximum and

minimum values. Therefore there is x e F such that

I f (x) | = | g(x) | = II f || p = || g II p . Thus pF is a real-

valued function, and Pp(f) ^ 0 for f e C(E; K).

Let X e K, F be a compact subset of E, and

f, g e C(E; K). Then p pUf) = || Xf || p =

sup{|Xf(x)[ |x e F} = sup{|X| |f(x)| |x e F}=

| X | sup { | f (x) | |x e F} = | X| I| f || F = |X|pp(f). Next

Pp(f + g) = j| f + g11 p = sup{ | f (X) + g(x) I |x £ F} <

sup{(|f(x)| + |gCx>|)|x e F} <

21

sup { | f (x) | |x e F} + sup { j g (x) | |x e F} = ||f||F + || g 11 p =

Pp(f) + PpCg). Thus pF(f + g) <_ Pp(f) + Pp(g). Then

pF(f*g) = || f * g I) p = sup{ | f (x)g(x) | |x e F> =

sup{(|f(x)| |g(x)|)|x e F} £

(sup{|f(x)| |x e F})(sup{|g(x)| |x e F}) =

II f II F II §11 p = pF(f)pF(g). Thus pp(f*g) < pF(f)pF(g).

Let h(x) = 1, where x e E. Since h is a continuous K-valued

function and h»g = g for every g e C(E; K), then C(E; K) has

a unit, and pF(h) = ||h||F = sup{|h(x)| |x e F} = 1. By

Definition 3.12, every nonempty compact subset F of E deter-

mines an algebra seminorm Pp = ||f||F on C(E; K). Thus the

theorem is proved.

Consider the collection of all B (f), where C»Pi

f e C(E; K), E is completely regular, £ > 0» and i e I.

The indexing set I will be the nonempty compact subsets of

E.

Theorem 3.16: Let E be a completely regular space.

Consider the algebra seminorm Pp defined on C(E; K), where

Pp(f) = II f 11F» f e C(E; K) , and F is a compact subset of

E. Let T = be "the family of all such algebra semi-

norms. Then T = is a directed family of seminorms.

Proof: Let il5 i2 e I and f e C(E; K). Then

p^ (f) = ||f||p, where F is a compact subset of E, and

p- (f) = ||f||o j where S is a compact subset of E. By 2 b

22

Theorem 1.26, F 0 S is compact. Let i = F U S. Then

p^(f) = IIf II gyp* L e t ^ = 1* T h e n — Xp^(f)>and

p.^Cf) <_ Xp^(f). Thus T is a directed family of seminorms

by Definition 3.9.

Definition 3.17: Let T = be the directed family

of algebra seminorms defined in Theorem 3.16. The natural

topology Tp on C(E; K) is the topology defined as in Defi-

nition 3.4 by r.

Theorem 3.18: The algebra C(E; K) is a topological

algebra with respect to Tp.

Proof: The proof of this theorem follows directly from

Theorem 3.13.

Theorem 3.19: Let T = "the directed family of

algebra seminorms defined in Theorem 3.16. Let f e C(E; K).

The collection of all Br (f), where £ > 0 , i e l , is a S 5Pi

basis of open neighborhoods at f with respect to Tp.

Proof: The proof of this theorem follows directly from

Theorem 3.10.

Definition 3.20: Let E be a completely regular space.

A subset L C C(E; R) is a lattice if f, g e L imply

sup(f, g) e L and inf (f, g) e L.

Lemma 3.21: Let X be a compact Hausdorff space,

L C C(X; R) a lattice!, and f e C(X; R). If for any Xj , x2 e X

and ^ > 0 there is a g e L such that |g(xx) - fCx^l < £

and |g(x2) - f(x2)| < £, then there exists h e L such that

|| f - h 11 x c £.

23

Proof: Let f e C(X; R) and £ > 0. It is necessary to

show that given t e X there is a e L such that

g^Cu) > f(u) - £ for any u e X and g^(t) < f(t) + £. Let

t e X. For any x e X,there is gv e L such that A.

|gx(t) - f(t)| < £ and |gx(x) - f(x)| < £. Let

Vx = {u E X| gx(u) > f(u) - £}. Since g - f is continuous,

V is open. Also V contains x. Since X is a compact space, A. A

there are x. , . . . , x„ e X such that V II . . . IJ Vv = X. i n Xj » w

Let g. = sup(g , . . . , g ) E L. If u E X, there is an i X1

such that u e V „ , 1 < i < n. Thus i - -

gt(u) > g (u) > f(u) - £. Since gv (t) < f(t) + £ for xi i

every i = 1, . . . , n, then g^(t) < f(t) + £. Thus given

t e X, there is a g^ e L such that g^Cu) > f(u) - £ for any

u e X and < f(t) + £.

For every t E X, choose g^ E L such that g^(u) > f(u) - £

for any u £ X and g^Ct) < f(t) + £. Set

= {u £ X| gt(u) < f(u) + £}. Since g and f are con-

tinuous, is open. Also contains t. Since X is com-

pact , there are t x, . . . , t £ X such that

Vt U • • • 0 V-f. = X. Set h = inf(gt g^ ) e L. 1 n 1 n

If u e X, there is an i such that u e V. ; thus i

h(u) < g+ (u) < f(u) + £. Moreover, since — Li g^-.Cu) > f(u) - E, for every i = 1, . . . , n, then Ti

h(u) > f(u) - £ for u £ X. Thus |h(u) - f(u)| < £ for

every u e X. Therefore sup{|h(u) - f(u)| |u £ X} < £, and

||h - f|[x < S .

24

Theorem 3.22 (Kakutani-Stone): Let E be a completely

regular space, L G C(E; R) a lattice, and f e CCE; R). Then

f belongs to the closure of L in C(E; R) if and only if for

any Xj , x2 e E and E, > 0 there is g e L such that

IgCx^ - f(xx)| < 5 and |g(x2) -f(x2)| <

Proof: Assume f e L. If f e L, then the theorem is

proved. Suppose f t L. Then f e L'. Let xx, x2 e E and

5 > 0. Let K be the compact subset of E consisting of Xi

and x 2 • Let i = K. Since B (f) = {g| ||g - f|| < is £ »Pi &

an open set containing f and f e L', there exists h e L

such that ||h - f||K < ?• Thus jhCx^ - f Cxx) | < £, and

IhCx^) - f(x2)| < Therefore the necessity of the condi-

tion is proved.

Next it is necessary to prove sufficiency. Let f

satisfy the conditions stated in the theorem. If f e L,

then the theorem is proved. Suppose f i L. Then it is

necessary to show that f e L'. Consider the collection of

all B_ (f), where £ > 0, i e I. This collection is a

basis of open neighborhoods at f with respect to T ,. In

order to show that f e L', it is necessary to show that for

any E, > 0 and any compact subset K of E, there exists h e L

such that ||h - f||j < £• Let £ > 0 and K be a compact sub-

set of E. Since f: E •+ R is continuous, (f|K): K + R is con-

tinuous. Let y e L C C(E; R). Then (y|K) e C(K; R). Set

Lj = {(y|K)| y e L}. It will be shown that Lx C C(K; R) is

a lattice. Suppose (p|K), (q|K) e Lx. By the definition of

25

a lattice, t = sup(p, q) e L. If x e K, then

t(x) = sup(p(x), q(x)) = sup((p|K)(x), (q|K)(x)). Then

(t|K) = sup((pjK), (q|K)) e Li. Similarly

inf(p|k, q|k) e L2. Thus Li is a lattice. For any

xz, x2 e E and £ > 0, there is g e L such that

IgCxj) - f(xx)| < E, and |g(x2) - f(x2)| < ?. Let xx, x2 e K

and ^ > 0. Then there is a g e L such that

jg(xi) - f(xi)J < £ and |g(x2) - f(x2)| < £. Since

(g|K) e Li, (g|K)(x1) = gCXj), (g|K)(x2) = g(x2),

(f|K)(Xj) = f(xx), and (f|K)(x2) = f(x2), then there exists

(g|K) e Lj such that |(g|K)(Xj) - (f|K)(x1)| < 5 and

|(g|K)(x2) - (f|K)(x )| < £. By Lemma 3.21, there exists

s e L such that || s - (f|K)|| < £. Since s = (h|K), * K

where h e Lx then ||h - f||K = ||s - (f|K)|| < £. Thus for ^ K

any £ > 0 and any compact subset K of E, there exists h e L

such that ||h - f || < Therefore f e L' C. L. Thus the K

theorem is proved.

CHAPTER IV

THE STONE-WEIERSTRASS THEOREM

The Stone-Weierstrass theorem is a generalization of the

Weierstrass approximation theorem. The Weierstrass approxi-

mation theorem shows that the set P of polynomials defined on

a closed interval [a, b] is dense in C[a, b], the continuous

real-valued functions defined on [a, b]. In order to gener-

alize the Weierstrass approximation theorem, replace [a, b]

with an arbitrary, completely regular space E and make a

similar statement concerning C(E; K). Since polynomials are

not necessarily defined on E, consider the set P. The set P

is an algebra of continuous real-valued functions defined on

[a, b]. Moreover, P is a subalgebra of C[a, b]. Let A be a

subalgebra of C(E; K), where E is completely regular. The

Stone-Weierstrass theorem shows that if A has certain speci-

fied properties which are also possessed by P, then A is

dense in C(E; K).

Definition 4.1: Let E be a completely regular space.

A subset X C C(E; C) is self-adjoint if f e X implies that

7 e X, where 7 is the complex-conjugate of f.

Theorem 4.2 (Stone-Weierstrass): Let E be a completely

regular space, A C C(E; K) a subalgebra which is assumed to

be self-adjoint in the complex case, and f e C(E; K). Then

26

27

f belongs to the closure of A in C(E; K) if and only if the

following conditions are satisfied:

(1) for any xx, x2 e E such that f(Xj) t f(x2), there

is g e A such that g(Xj) i g(x2); and

(2) for any x e E such that f(x) t 0, there is g e A

such that g(x) i 0.

The following lemmas are needed in the proof of the

Stone-Weierstrass theorem.

Lemma .3: If K > 0 and £ > 0, there is a polynomial

p: R R such that |p(t) - |t|| < £ for |t| _< K.

Proof: Let 5 > 0 and K >_ 0. The function f, where

f(t) = |t|, is continuous on the closed interval C-K, K].

By the Weierstrass approximation theorem, there exists a

polynomial p with the property that |p(t) - |t|| < £ for

|t| < K.

Lemma Let E be a completely regular space. Every

closed subalgebra A of C(E; R) is a lattice.

Proof: Let f e A. It will be shown that f e A implies

| f | e A. Let K C E be compact, and let S = ||f||„ • Let

E, > 0. By Lemma 4.3, there exists a polynomial p(f) such

that |p[f(x)] - |f(x)| | < £ for x e K. Since p(f) - |f| is

a real-valued continuous function defined on a compact set K,

then p(f) - J f| assumes its maximum and minimum values on K.

Thus ||p(f) - | f i || < £. Since A is an algebra, p(f) e A. K

Since f e C(E; R), |f| e C(E; R). The collection of all

Br (|f|), where £ > 0, i e I, is a basis of open sets at ^ >Pi

28

jf[. Thus for £ > 0 and i e I,there exists a polynomial

p(f) e A such that p(f) e Br (|f|). Suppose |f| t A. Then s> sP£

by Theorem 1.32,|f| e A = A. This contradiction proves that

|fj e A. It is now possible to show that A is-a lattice.

Let f, g e A. Then f - g , j f — g[ e A. Since

sup(f, g) = (f + g + |f - g|)/2 e A and inf(f, g) =

(f + g - |f - g|)/2 e A, A is a lattice.

Lemma M-. 5 : Let R2 be the cartesian square of R. If

(a, b), (c, d) e R2 and a e R, define (a, b) + (c, d) =

(a + c, b + d), a(a, b) = (aa, ab) , and (a, b)(c, d) =

(ac, bd). With respect to these operations, R2 is an algebra.

The subalgebras of R2 are R2, 0 x R, R x 0, 0, and A, where

A = {(x, y) e R2| x = y>.

Proof: It is clear that R2 is an algebra and that R2,

0 x R, R x 0, 0,and A are subalgebras of R2. Let A be a sub-

algebra of R2 distinct from R2, 0, R x 0, and 0 x R. Since

A i R and A t 0, then by Theorem 1.9, A is a vector subspace

of dimension one. By Theorem 1.9, A is a line in R2 contain-

ing (0, 0). Since A t R x 0 and A i 0 x R, then A is given

by an equation y = a x , a e R , a ^ 0 . Since (1, a) satisfies

the equation, then (1, a) e A. Since (1, a)2 e A and

(1, a)2 = (1, a2), then (1, a 2) satisfies the equation and

a2 = a. Thus a = 1. Therefore A = A. Thus the proof is

complete.

Corollary 4.6: Let A C R2 be a subalgebra and b e R2.

Then b t A if and only if at least one of the.following

29

conditions hold true: ( l ) b ^ R x O and A C R x 0;

(2) b t 0 x R and A £. 0 x R;

(3) b i A and A C A.

Proof: Each of (1), (2), and (3) implies b t A. Con-

versely assume b t A. Then A i R2, and b i 0. Suppose A = 0.

Then A C 0 x R, and A C R x o. Since b i 0, then either

b £ 0 x R o r b £ R x 0 . Thus (1) or (2) is satisfied. Sup-

pose A i- 0. Then A = R x o or A = 0 x R or A = A. Thus one

of the conditions (1), (2), or (3) will be satisfied. Thus

the theorem is proved.

Proof of Theorem M-. 2 in the real case: First the neces-

sity will be proved. Assume f e A. If f e A, then the

theorem is proved. Suppose f t A. Then f e A'. Let Xj,

x2 e E such that f(Xj) i f(x2). Let K be the compact subset

of E consisting of xx, x2. Let £ = |f(x2) - f(Xj)| > 0.

Since each w^ e r e i £ is an open set contain-

ing f, there exists g e A such that [} g - f|| < ?/2. Thus

|g(Xj) - f(Xj)| < £/2, and |g(x2) - f(x2)| < £/2. It is

necessary to show that g(xx) t g(x2). Suppose

g(xx) = g(x2). Then |f(xx) - f(x2)|

|g(xj) - f(xj.)| + |f(x2) - g(x2)| < 5 = | f (x i) - f(x2)|.

This contradiction proves that g(xj) i g(x2). Thus the

necessity of (1) is proved. Let x e E such that f(x) i 0,

and let £ = |f(x)| > 0. Since B = {x} is compact and each

Br Cf) is an open set containing f, there exists g e A S> sPj_ such that || g - f 11 < £. Thus |g(x) - f (x) | < £. It is

U

30

necessary to show that g(x) i 0. Suppose g(x) = 0. Then

|f(x)| < |f(x)|. This contradiction proves that g(x) t 0.

Thus the necessity of (2) is proved.

Now assume (1) and (2). Let R2 have the same algebraic

operations as in Lemma 4.5. Let , x2 e E and 5 > 0. It

will be shown that the mapping g e C(E; R) (gCx^, g(x2) e R2

is an algebra homomorphism. Call this mapping $. It has been

shown that the algebra C(E; R) is a vector space over R and

that the algebra R2 is a vector space over R. Let

g, h e C(E; R) and c e R. Then $(g + h) =

((g + h)(xx), (g + h)(x2)) = (g(Xl) + h(xj), g(x2) + h(x2)) =

(gCxj), g(x2)) + (h(xj), h(x2)) = $(g) + $(h). Also

$(c-g) = (cg(xx), cg(x2)) = c(g(Xj), g(x2)) = c$(g).

Next $(gh) = (g(x:)h(Xj) , g(x2)h(x2)) =

g(x2))(h(x1), h(x2)) = $(g)<Kh). Thus $ is an

algebra homomorphism. By Theorem 1.13, 4>(A) is a subalgebra

of R2. Suppose $(f) = (f(Xj), f(x2)) t R x o. Then

f(x2) i 0. By (2), there exists g e A such that g(x2) i 0.

Thus $(A) <£ R x 0. Similarly if $(f) t 0 x R, then

<KA) <£ 0 x R. Suppose $(f) = (f(xx), f(x2)) I A. Then

f(Xj) t f(x2). By (1), there exists g e A such that

g(Xj) i g(x2). Thus $(A) (p A. Thus by Corollary 4.6,

$(f) e $(A). Therefore there exists g e A such that

f(xx) = gCXj) and f(x2) = g(x2). Since g e A C. ~k and since

A is a lattice by Lemma 4.4, then by the Kakutani-Stone theo-

rem f e A = A. Thus the theorem is proved.

31

Lemma 4.7:. Let E be a completely regular space and

A C C(E; C) a self-adjoint algebra. Then the set RA of the

real parts Rf of all f e A is a subalgebra of C(E; R), and

A = RA + iRA.

Proof: Let h e RA. Then there exists g e A, where

g = s + it and s, t e C(E; R), such that h = Rg. Then

h = Rg = g * S = s e C(E; R). Thus RA C C(E; R). It is

clear that RA is a vector subspace of C(E; R). Let f e A.

if

Since A is a self-adjoint algebra, then Rf = — - — e A.

Thus RAC A. If f, g e RA, then f, g e A. Thus fg e A.

Since fg is a continuous real-valued function, fg = R(fg) eRA.

Thus RA is a subalgebra of C(E; R).

Since A is a vector space over C and RA C A, then

RA + iRA C A. If f e A, then f = fx + if2, where

f1, f2 e C(E; R). Thus fx = Rf e RA,and f2 = R(-if) £ RA.

Then f e RA + iRA. Therefore A = RA + iRA.

Lemma 4.8: The mapping (gj, g2) e C(E; R) x C(E; R) -*

gj + ig2 e C(E; C) is a homeomorphism.

Proof: Call this mapping $. It is clear that $ is a

one-to-one, onto mapping. It will be shown that $ is con-

tinuous by using Theorem 1.37. Let Cg:, g2) e C(E; R) xC(E;R),

and let <Kg,, g„) = g. The collection of all B (g)> where 1 2 5»Pi

£ > 0, i e I, is a subbasis of open neighborhoods at g. Let

£ > 0 and i e I. It is necessary to show that there exists

an open set 0 in C(E; R) x C(E; R) such that (gi> g 2 e 0 a n d C L e t x £ B£/2, p.^Si) a n d

32

y £ B5/2,pi<g'K T h e n <X' y ) E W , ' 8 . ' * BC/2, Pi < g^

which is open in C(E; R) x C(E; R). It will be shown that

f(x, y) = x + iy e B (g)« Let i = K, where K is compact. S s P-L

Then ||g1 + ig2 - (x + ±y) ||K = || (gx - x) + (g2 -y>i|lK <

Us, - x||K + II Cgs - y)i|lK = IIs, - x|lK • IIs2 - yIIK <

£/2 + £/2 = g. Thus x + iy e B (g). Therefore $ is con-£»Pi

tinuous.

Next it will be shown that $-1 is continuous by using

Theorem 1.37. Let g = gx + ig2 e C(E; C). Then

$_1(g) = (gi, g2). Let A = -C 01 x 02 | gx e 0l9 g2 e 02> and

Oi and 02 are open subsets of C(E; R)}. Then A is a basis

a-t (gi s Sz)' Let B = {C(E; R) x 021 g2 £ 0^ and 02 is an

open subset of C(E; R)}, and let C = {Oi x C(E; R)| gx e 0X

and is an open subset of C(E; R). By Theorem 1.22 and

Definition 1.21, F = B U C is a subbasis of open neighbor-

hoods at (gj, g2) with respect to the basis A. Let

D = C(E; R) x G e F, where g2 e G, G is open in C(E; R).

There exists an open set B^ p.(g2) C G, where E, > 0, i e I.

It is necessary to show that there exists an open subset V

of C(E; C) such that g e V and $_I(V) C C(E; R) x B (g2)C £ »Pi

C(E; R) x G. Let V = Br _ (g),and let s = s, + is, e V. It S 5P1 A *

will be shown that $~:(s) = (s,, s ) e C(E; R) x B (g2)« ?>Pi

Let i = K, where K is a compact subset of E. Now

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Hg2 " S 2"k = SUP{IS2(

X> " sa(x)||x e K} =

sup{"\/(g (x) - s (x) )* I x e K} < 2 2 1 —

sup{V(g2 (x) - s2(x))2 + (gi(x) - s1(x))2| x e K} =

sup{|g1(x) + ig2(x) - s:(x) - is2(x)||x e K} =

llgi + ig, ~ s - is || < g. Thus s e B (g ). 1 2 1 2 K 2 2

Clearly s, e C(E; R). Thus f_1(V) C C(E; R) x G. If

P = S x C(E; R) e F, where S is open in C(E; R) and

gj e S, had been chosen instead of D = C(E; R) x G, then

a similar proof would yield the desired result. Therefore

S*-1 is continuous at g by Theorem 1.37, and $ is a homeo-

morphism.

Lemma ^. 9: For every X, Y CC(E; R), X + iY = X + iY.

Proof: By Lemma 4.8, the mapping $ is a homeomorphism.

Let X, Y C C(E; R). By Theorem 1.29, X x Y = X x Y. Since

X x y is closed, X x Y is closed. Since $ is a homeomor-

phism, $(X, Y) = X + iY is closed. Then X + iY =

X + iY O X + iY. It will be shown that X + iY C X + iY.

Suppose there exists f = g + ihe3( + i7 such that

f £ X + iY. Then there exists an open set 0 in C(E; C)

such that f e 0 and 0 does not contain an element of X + iY.

Since $ is a homeomorphism, $""1(0) is open, and (g, h) e$~1(0)i

There exists a basic element 01 x 02 of the product topology

on C(E; R) x C(E; R) such that (g, h) e Oj x 02 C $-1(0).

Then g e 0 , and h e 02, where 0X and 02 are open subsets of

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C(E; R). Since g e X, h e 7, 0^ is open,and 0^ is open, then

0X contains a point z of X and 02 contains a point w of Y.

Then z + iw e fCOj x 0 2 ) C $$_1(0) C 0, where

z + iw e X + iY. This contradiction proves that

X + iYC X + iY. Therefore X + iY = X + iY.

Proof of Theorem 4.2 in the complex case: Let

f e C(E; C) and A be a subalgebra of C(E; C). Necessity is

proved as in the real case. Conversely, assume (1) and (2).

Write f = fj + if2, where f x , f2 e C(E; R). Using the re-

sults of Theorem 4.2 as shown in the real case, it will be

shown that f e RA and f2 e RA. By Lemma 4.7, RA is a sub-

algebra of C(E; R). Let xx, x£ e E such that f^x ) i f (x ).

Then f(Xj) i f(x2). Thus there is g e A such that

g(xx) i g(x2). Write g = gx + ig2, where g1, g2 e C(E; R).

Then gj = Rg e RA and g2 = R(-ig) e RA. Since g(Xj) i g(x2),

then either gj(Xj) t gjCXg) or g2(x1) i g2(x2). Thus fj and

RA satisfy (1) of Theorem 4.2 in the real case. Let x e E

such that fx(x) i 0. Then f(x) i 0. There exists g e A

such that g(x) i 0. Write g = gx + ig2, where

g , g2 e C(E; R). Since g(x) i 0, either gx(x) / 0 or

g2(x) i 0. Clearly gx, g2 e RA. Thus f and RA satisfy

(2) of Theorem 4.2 in the real case. Therefore fx e RA.

Also f. e KK. By Lemma 4.7 and Lemma 4.9,f = f, + if e 2 ' 1 2

RA + iRA = RA + iRA = A.

The following corollary to the Stone-Weierstrass theorem

demonstrates more clearly the relationship between the Weier-

strass approximation theorem and the Stone-Weierstrass theorem.

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Corollary M-. 10 : Let E be a completely regular space and

A C C(E; K) be a subalgebra which is assumed to be self-

adjoint in the complex case. Then A is dense in C(E; K)

if and only if the following conditions are satisfied:

(1) for any x , x e E such that x t x .there is 1 2 1 2

g e A such that g(x:) i g(x^); and

(2) for any x e E,there is g e A such that g(x) t 0.

Proof: Assume A = C(E; K). Let , x2 e E, xl i x2.

Since E is a Hausdorff space, points are closed subsets of E.

Since E is completely regular, there is f e C(E; K) = "K such

that f(xx) i f(x2). By Theorem U.2, there is g e A such that

g(Xj) i g(x2). Thus (1) is satisfied. If x e E, there is

f e C(E; K) = A, such as f = 1, where f(x) ^0. By Theorem

4.2,there is g e A such that g(x) t 0. Thus (2) is satisfied.

Conversely, assume (1) and (2) are true. Let

f e C(E; K). Then (1) and (2) of Theorem 4.2 are satisfied

for f. Thus f e A. Therefore X = C(E; K).

BIBLIOGRAPHY

Nachbin, Leopoldo, Elements of Approximation Theory, Princeton, N. J. , D. Van Nostrand Company, Inc., 1967.

Paige, Lowell J. and J. Dean Swift, Elements of Linear Algebra, Waltham, Mass., Blaisdell Publishing Company, 1961.

Royden, H. L., Real Analysis, New York, N.Y., The Mac-millan Company, 1969.

Simmons, George F., Introduction to Topology and Modern Analysis, New York, N.Y., McGraw-Hill Book Company, Inc. , 1963 .

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