Weierstrasstételben

Weierstraß's theorem, also known as the Stone-Weierstraß theorem, is a fundamental result in real and complex analysis. In its simplest form, it states that if a continuous real-valued function is defined on a compact set, then it can be uniformly approximated by a polynomial function. More generally, the theorem concerns the density of certain subalgebras of continuous functions within the space of all continuous functions on a compact Hausdorff space.

Specifically, let X be a compact Hausdorff space and C(X) be the space of all continuous real-valued functions on X. A subalgebra A of C(X) is said to be a separating subalgebra if for any two distinct points x and y in X, there exists a function f in A such that f(x) is not equal to f(y). The Stone-Weierstraß theorem states that if A is a separating subalgebra of C(X) that contains the constant functions, then A is dense in C(X) with respect to the uniform norm. This means that for any function g in C(X) and any arbitrarily small positive number epsilon, there exists a function f in A such that the maximum absolute difference between f(x) and g(x) over all x in X is less than epsilon.

The theorem has important applications in various areas of mathematics, including approximation theory, functional analysis, and differential geometry. It guarantees the existence of good approximations for complicated functions by simpler ones, which is a powerful tool in theoretical and applied mathematics. The complex version of the theorem also exists, with a slight modification: the subalgebra must be closed under complex conjugation.