StoneWeierstrasstétel

The Stone-Weierstrass theorem is a fundamental result in real analysis and functional analysis. It states that if you have a space of continuous real-valued functions on a compact Hausdorff space, and this space is a subalgebra that separates points, then it is dense in the space of all continuous real-valued functions on that space, with respect to the uniform norm.

A subalgebra of continuous functions means that if you take any two functions from the set, their sum and product are also in the set. Separating points means that for any two distinct points in the space, there exists a function in the set that takes different values at those two points. The uniform norm, also known as the supremum norm, measures the maximum difference between two functions over the entire space.

In simpler terms, the theorem guarantees that you can approximate any continuous function on a closed and bounded set of real numbers arbitrarily well using polynomials. This is a generalization of the Weierstrass approximation theorem, which specifically deals with polynomial approximations. The Stone-Weierstrass theorem has broad implications in various fields of mathematics and its applications, providing a powerful tool for approximating complex functions with simpler ones. It is crucial in areas like numerical analysis, approximation theory, and the study of function spaces.