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 \( X \).
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A point-separating real subalgebra of that contains the constants is uniformly dense when 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.