Stone-Weierstrass theorem

The Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be uniformly approximated as closely as desired by a polynomial function. Because polynomials are the simplest functions, and computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance.

Marshall H. Stone considerably generalized the theorem and simplified the proof; his result is known as the Stone-Weierstrass theorem. The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the compact interval [a,b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, approximation with elements from other subalgebras of C(X) is investigated.

The crucial property that a subalgebra must satisfy is that it separates points: A subset A of C(X) is said to separate points if, for every two different points x and y in X and every two real numbers a and b, there exists a function p in A with p(x) = a and p(y) = b. Note that a sometimes the term separates points is given the slightly weaker meaning that for every two different points x and y in X there exists a function p in A with p(x) not equal to p(y). The theorem is true with the weaker meaning of separation of points.

Table of contents

1 Stone-Weierstrass theorem 2 Stone-Weierstrass theorem, real algebra version 3 Stone-Weierstrass theorem, lattice version 3.1 Applications 3.1.1 Weierstrass approximation theorem 3.1.2 Other consequences

Stone-Weierstrass theorem

Let K be a compact Hausdorff space and denote by C(K) the space of continuous complex functions on K with the topology of uniform convergence. Suppose that S is a set of continuous complex functions on K which separates points of K. Then, the complex unital *-algebra generated by S is dense in C(K).

Stone-Weierstrass theorem, real algebra version

If X is a compact Hausdorff space with at least two points and A is a subalgebra of the Banach algebra C(X) which separates points and contains a non-zero constant function, then A is dense in C(X).

This implies Weierstrass' original statement since the polynomials on [a,b] form a subalgebra of C[a,b] which separates points.

Stone-Weierstrass theorem, lattice version

Let X be a compact Hausdorff space. A subset L of C(X) is called a lattice in C(X) if for any two elements f, g in L, the functions max(f,g) and min(f,g) also belong to L. The lattice version of the Stone-Weierstrass theorem states:

If X is a compact Hausdorff space with at least two points and L is a lattice in C(X) which separates points, then A is dense in C(X).

Applications

As has been pointed out, the Weierstrass approximation theorem is a straightforward application of the Stone-Weierstrass theorem.

Weierstrass approximation theorem

Suppose f is a continuous function defined on the interval [a,b] with real values. For every ε>0, there exists a polynomial function p with real coefficients such that for all x in [a,b], we have |f(x) - p(x)| < ε.

The set C[a,b] of continuous real-valued functions on [a,b], together with the supremum norm ||f|| = sup_{x in [a,b]} |f(x)|, is a Banach algebra, (i.e. an associative algebra and a Banach space such that ||fg|| ≤ ||f|| ||g|| for all f, g). The set of all polynomial functions forms a subalgebra of C[a,b], and the content of the Weierstrass approximation theorem is that this subalgebra is dense in C[a,b].

Other consequences

The Stone-Weierstrass theorem can be used to prove the following two statements:

• If f is a continuous real-valued function defined on the set [a,b] x [c,d] and ε>0, then there exists a polynomial function p in two variables such that |f(x,y) - p(x,y)| < ε for all x in [a,b] and y in [c,d].
• If X and Y are two compact Hausdorff spaces and f : XxY -> R is a continuous function, then for every ε>0 there exist n>0 and continuous functions f₁, f₂, ..., f_n on X and continuous functions g₁, g₂, ..., g_n on Y such that ||f - ∑f_ig_i|| < ε