Kenneth Kuttler

202 CHAPTER 9. WEIERSTRASS APPROXIMATION THEOREM

Then D is a closed subspace of C(

X̃)

. For f ∈C0 (X) ,

f̃ (x)≡{

f (x) if x ∈ X0 if x = ∞

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. This proves the lemma.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 9.2.9 Let A be an algebra of functions in C0 (X ;R) where (X ,τ) is a locallycompact Hausdorff space which separates the points and annihilates no point. Then A isdense in C0 (X ;R).

Proof: Let(

X̃ , τ̃)

be the one point compactification as described in Lemma 7.12.20.

Let Ã denote all finite linear combinations of the form{n

∑i=1

ci f̃i + c0 : f ∈A , ci ∈ R

}

where for f ∈C0 (X ;R) ,

f̃ (x)≡{

f (x) if x ∈ X0 if x = ∞

Then Ã is obviously an algebra of functions in C(

X̃ ;R)

. It separates points because this istrue of A . Similarly, it annihilates no point because of the inclusion of c0 an arbitrary ele-ment of R in the definition above. Therefore from Theorem 9.2.5, Ã is dense in C

(X̃ ;R

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

X̃ ;R)

and so there exists a sequence {hn} ⊆ Ã

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 . This proves the theorem.

9.2.3 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 f ∈A .Such an algebra is called self adjoint.

202 CHAPTER 9. WEIERSTRASS APPROXIMATION THEOREMThen D is a closed subspace of C (2). For f € Co (X),Fo={ diteemand let 6 : Co (X) > D be given by 0f = f. Then @ is one to one and onto and also satisfies\|fllo = ||@f||... Now D is complete because it is a closed subspace of a complete spaceand so Co (X) with ||-||,, is also complete. This proves the lemma.The 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 Cp (X) all have real values, I will denote the resultingspace by Cy (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 9.2.9 Let & be an algebra of functions in Co(X;IR) where (X,7) is a locallycompact Hausdorff space which separates the points and annihilates no point. Then & isdense in Coy (X;R).Proof: Let (x 7) be the one point compactification as described in Lemma 7.12.20.Let Dd denote all finite linear combinations of the formn ~Ycifitco: fed, cq ERi=lwhere for f € Co (X;R),Fo={ diteemThen / is obviously an algebra of functions in C (x ; R) . It separates points because this istrue of <. Similarly, it annihilates no point because of the inclusion of cg an arbitrary ele-ment of R in the definition above. Therefore from Theorem 9.2.5, of is dense in C (x ; R) .Letting f € Cy (X;R), it follows f EC (x R) and so there exists a sequence {h,} C asuch that h, converges uniformly to f. Now hy is of the form Yi_, c? fi + cp and sincef (ce) = 0, you can take each cj = 0 and so this has shown the existence of a sequence offunctions in .& such that it converges uniformly to f. This proves the theorem.9.2.3. 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 IR? The following is the version of the Stone Weierstrasstheorem which applies to this case. You have to assume that for f € .& it follows f € .&.Such an algebra is called self adjoint.