510 CHAPTER 19. HAUSDORFF SPACES AND MEASURES

and let θ : C0 (X)→D be given by θ f = f̃ . Then θ is one to one and onto and also satisfies∥ f∥

∞= ∥θ f∥

∞. Now D is complete because it is a closed subspace of a complete space

and so C0 (X) with ∥·∥∞

is also complete. ■The above refers to functions which have values in C but the same proof works for

functions which have values in any complete normed linear space.In the case where the functions in C0 (X) all have real values, I will denote the resulting

space by C0 (X ;R) with similar meanings in other cases.With this lemma, the generalization of the Stone Weierstrass theorem to locally com-

pact sets is as follows.

Theorem 19.4.3 Let A be an algebra of functions in C0 (X ;R) where (X ,τ) is alocally compact Hausdorff space which separates the points and annihilates no point. ThenA is dense in C0 (X ;R).

Proof: Let(

X̃ , τ̃)

be the one point compactification as described in Lemma 19.1.27.

Let à denote all finite linear combinations,{

∑ni=1 ci f̃i + c0 : f ∈A , ci ∈ R

}where for

f ∈C0 (X ;R) ,

f̃ (x)≡{

f (x) if x ∈ X0 if x = ∞

.

Then à is obviously an algebra of functions in C(

X̃ ;R)

. It separates points because thisis true of A . Similarly, it annihilates no point because of the inclusion of c0 an arbitraryelement of R in the definition above. Therefore from Theorem 5.10.5, Ã is dense inC(

X̃ ;R). Letting f ∈C0 (X ;R) , it follows f̃ ∈C

(X̃ ;R

)so there exists a sequence {hn}⊆

à such that hn converges uniformly to f̃ . Now hn is of the form ∑ni=1 cn

i f̃ ni + cn

0 and sincef̃ (∞) = 0, you can take each cn

0 = 0 and so this has shown the existence of a sequence offunctions in A such that it converges uniformly to f . ■

19.4.2 The Case of Complex Valued FunctionsWhat about the general case where C0 (X) consists of complex valued functions and thefield of scalars is C rather than R? The following is the version of the Stone Weierstrasstheorem which applies to this case. You have to assume that for f ∈A it follows f̄ ∈A .

Lemma 19.4.4 Let z be a complex number. Then Re(z) = Im(i z̄) , Im(z) = Re(i z̄) .

Proof: The following computation comes from the definition of real and imaginaryparts.

Re(z) =z+ z̄

2=

iz+ i z̄2i

=i z̄− (i z̄)

2i= Im(i z̄)

Im(z) =z− z̄

2i=

i z̄− iz2

=i z̄+(i z̄)

2= Re(i z̄) ■

Theorem 19.4.5 Suppose A is an algebra of functions in C0 (X) for X a locallycompact Hausdorff space which separates the points of X and annihilates no point of X ,and has the property that if f ∈A , then f̄ ∈A . Then A is dense in C0 (X).

510 CHAPTER 19. HAUSDORFF SPACES AND MEASURESand let 0 : Co (X) > D be given by Of = f. Then @ is one to one and onto and also satisfieslf llc. = ||@||... Now D is complete because it is a closed subspace of a complete spaceand so Co (X) with ||-||,, is also complete. HlThe above refers to functions which have values in C but the same proof works forfunctions which have values in any complete normed linear space.In the case where the functions in Co (X) all have real values, I will denote the resultingspace by Cp (X;R) with similar meanings in other cases.With this lemma, the generalization of the Stone Weierstrass theorem to locally com-pact sets is as follows.Theorem 19.4.3 Let. be an algebra of functions in Co (X;R) where (X,7) is alocally compact Hausdorff space which separates the points and annihilates no point. Thenfl is dense in Co (X;R).Proof: Let (x 7) be the one point compactification as described in Lemma 19.1.27.Let / denote all finite linear combinations, {rn cif; tco fED, GE R} where forf€Co(X;R),=) —_ J f(x) ifxexFW) = rf 0 if x= 0Then / is obviously an algebra of functions in C (x R), It separates points because thisis true of </. Similarly, it annihilates no point because of the inclusion of co an arbitraryelement of R in the definition above. Therefore from Theorem 5.10.5, & is dense inC (x :R) . Letting f € Co (X;R), it follows f €C (x :R) so there exists a sequence {h, } Cof such that hy converges uniformly to f. Now h,, is of the form Y_, c? f? + cg and sincef () = 0, you can take each co = 0 and so this has shown the existence of a sequence offunctions in . such that it converges uniformly to f. Hi19.4.2 The Case of Complex Valued FunctionsWhat about the general case where Cp (X) consists of complex valued functions and thefield of scalars is C rather than R? The following is the version of the Stone Weierstrasstheorem which applies to this case. You have to assume that for f € / it follows f € a.Lemma 19.4.4 Let z be a complex number. Then Re(z) = Im (iz), Im(z) = Re (iz).Proof: The following computation comes from the definition of real and imaginaryparts.z+ igtiz — iz—(iz)Rez) = 7 5, = Im(i2)—2 iz-ig iz+(i2 _Im(z) = == S f=+ 3) _ Re (iz) aTheorem 19.4.5 Suppose & is an algebra of functions in Co(X) for X a locallycompact Hausdorff space which separates the points of X and annihilates no point of X,and has the property that if f € &, then f € &. Then &f is dense in Co(X).