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).