 Stone–Weierstrass theorem

A point-separating real subalgebra of $C(X)$ that contains the constants is uniformly dense when $X$ is compact Hausdorff. Polynomial approximation of the square root makes its closure a lattice; finite maxima and minima then turn pointwise interpolation into uniform approximation.

The Stone–Weierstrass theorem is a fundamental result in analysis that provides conditions under which a set of functions can approximate continuous functions on a compact space. It generalizes the Weierstrass approximation theorem, which specifically addresses polynomial functions. Here is a more formal statement of the theorem: Let \( X \) be a compact Hausdorff space, and let \( C(X) \) denote the space of continuous real-valued functions on \( X \).