3.2. THE CASE OF COMPACT SETS 81

3.2 The Case of Compact SetsThere is a profound generalization of the Weierstrass approximation theorem due to Stone.It has to be one of the most elegant and insightful theorems in mathematics.

Definition 3.2.1 A is an algebra of real valued functions if A is a real vector spaceand if whenever f ,g ∈A then f g ∈A .

There is a generalization of the Weierstrass theorem due to Stone in which an intervalwill be replaced by a compact or locally compact set and polynomials will be replaced withelements of an algebra satisfying certain axioms.

Corollary 3.2.2 On the interval [−M,M], there exist polynomials pn such that

pn (0) = 0

andlimn→∞∥pn−|·∥|∞ = 0.

recall that ∥ f∥∞≡ supt∈[−M,M] | f (t)|.

Proof: By Corollary 3.1.3 there exists a sequence of polynomials, {p̃n} such that p̃n→|·| uniformly. Then let pn (t)≡ p̃n (t)− p̃n (0) . ■

In what follows, x will be a point in Rp. However, this could be generalized. Note thatCp can be considered as R2p.

Definition 3.2.3 An algebra of functions A defined on A, annihilates no point ofA if for all x ∈ A, there exists g ∈ A such that g(x) ̸= 0. The algebra separates points ifwhenever x1 ̸= x2, then there exists g ∈A such that g(x1) ̸= g(x2).

The following generalization is known as the Stone Weierstrass approximation theorem.

Theorem 3.2.4 Let A be a compact set in Rp and let A ⊆C (A;R) be an algebra offunctions which separates points and annihilates no point. Then A is dense in C (A;R).

Proof: First here is a lemma.

Lemma 3.2.5 Let c1 and c2 be two real numbers and let x1 ̸= x2 be two points of A.Then there exists a function fx1x2 such that

fx1x2 (x1) = c1, fx1x2 (x2) = c2.

Proof of the lemma: Let g ∈ A satisfy g(x1) ̸= g(x2). Such a g exists because thealgebra separates points. Since the algebra annihilates no point, there exist functions h andk such that h(x1) ̸= 0, k (x2) ̸= 0. Then let

u≡ gh−g(x2)h, v≡ gk−g(x1)k.

It follows that u(x1) ̸= 0 and u(x2) = 0 while v(x2) ̸= 0 and v(x1) = 0. Let

fx1x2 ≡c1u

u(x1)+

c2vv(x2)

.