Stone–Weierstrass theorem

The Stone–Weierstrass theorem gives conditions under which an algebra of continuous functions on a compact space is dense in the space of all continuous functions, typically with the uniform norm.

Compact space: A topological space in which every open cover has a finite subcover. Compactness supplies the domain condition in the theorem’s standard form.
Polynomial approximation: The approximation of functions by polynomials, often measured by uniform error on a specified set. The theorem extends polynomial density from intervals to algebras on compact spaces.
Marshall H. Stone: An American mathematician whose work established a broad approximation theorem for function algebras. Stone proved the general theorem that bears his name.
Fourier approximation: Approximation of periodic functions by finite sums of sines, cosines, or complex exponentials. Trigonometric polynomials form a dense algebra for continuous functions on the circle.
Müntz–Szász theorem: A criterion describing when selected powers of a variable are dense in continuous functions on an interval. It gives a specialized density test where Stone–Weierstrass supplies a broader algebraic criterion.
Continuous function: A function between topological spaces for which inverse images of open sets are open. The theorem compares an algebra of these functions with the full space of them.
Polynomial algebra: The algebra of polynomial functions generated by a chosen collection of functions and constants. The proof builds approximants from separating functions using sums, products, and constants.
Karl Weierstrass: A German mathematician known for foundational work in analysis, including polynomial approximation of continuous functions. The theorem generalizes the classical approximation result associated with Weierstrass.
Trigonometric polynomial: A finite linear combination of sine and cosine functions, or complex exponentials, with integer frequencies. These functions provide a concrete Stone–Weierstrass approximation family on compact groups such as the circle.
Runge's theorem: A complex-analysis theorem characterizing rational approximation on compact subsets of the complex plane. It yields approximation under geometric conditions rather than point separation in a function algebra.

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