
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 realvalued 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].
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 of the theorem is:
This generalizes Weierstrass' statement since the polynomials on [a,b] form a subalgebra of C[a,b] which separates points.
Note that the above theorem is also true if we replace the assumption that A separate points with the slightly weaker assumption 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 StoneWeierstrass 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 StoneWeierstrass theorem states:
Search Encyclopedia

Featured Article
