Because polynomials are the simplest functions, and computers can directly evaluate polynomials with exact precision, this theorem has both practical and theoretical relevance. [Marshall H. Stone]? considerably generalized the theorem and simplified the proof into the Stone-Weierstrass theorem. |
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 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].
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:
This generalizes Weierstrass' statement since the polynomials form a subalgebra which separates points.
The Stone-Weierstrass theorem can be used to prove the following two statements:
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: