Stone-Weierstrass theorem

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The Weierstrass approximation theorem states that every continuous function defined on an interval [a,b] can be 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.

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|| = supx 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].

Stone-Weierstrass theorem, algebra version

The approximation theorem is generalized 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 of the subalgebra 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. The formal statement is:

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 generalizes Weierstrass' statement since the polynomials form a subalgebra which separates points.

Applications

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, ..., fn on X and continuous functions g1, g2, ..., gn on Y such that ||f - ∑figi|| < ε

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).



Need to cover the case of complex valued functions.