Stone–Weierstrass theorem

In <a href="/facts/Mathematical_analysis/XXeR26Ih">mathematical analysis</a>, the Weierstrass approximation theorem states that every <a href="/facts/Continuous_function/sKbl02pB">continuous function</a> defined on a closed <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> [a, b] can be <a href="/facts/Uniform_convergence/ecz9eNWn">uniformly approximated</a> as closely as desired by a <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in <a href="/facts/Polynomial_interpolation/DmnuxaKT">polynomial interpolation</a>. The original version of this result was established by <a href="/facts/Karl_Weierstrass/4U8vdQzL">Karl Weierstrass</a> in 1885 using the <a href="/facts/Weierstrass_transform/1LNo4Bce">Weierstrass transform</a>.
<a href="/facts/Marshall_H._Stone/TFzbE4GH">Marshall H. Stone</a> considerably generalized the theorem and simplified the proof. His result is known as the Stone–Weierstrass theorem. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary <a href="/facts/Compact_space/d0cgXJH7">compact</a> <a href="/facts/Hausdorff_space/gbKIrusl">Hausdorff space</a> X is considered, and instead of the <a href="/facts/Algebra_over_a_field/8T8ulWJb">algebra</a> of polynomial functions, a variety of other families of continuous functions on 
 
 
 
 X
 
 
 {\displaystyle X}
 
 are shown to suffice, as is detailed below. The Stone–Weierstrass theorem is a vital result in the study of the algebra of <a href="/facts/Continuous_functions_on_a_compact_Hausdorff_space/v5Yf5tlM">continuous functions on a compact Hausdorff space</a>.
Further, there is a generalization of the Stone–Weierstrass theorem to noncompact <a href="/facts/Tychonoff_space/jyCgpKB5">Tychonoff spaces</a>, namely, any continuous function on a Tychonoff space is approximated <a href="/facts/Compact-open_topology/qMGsW8gk">uniformly on compact sets</a> by algebras of the type appearing in the Stone–Weierstrass theorem and described below.
A different generalization of Weierstrass' original theorem is <a href="/facts/Mergelyan%2527s_theorem/V9ErA7q8">Mergelyan's theorem</a>, which generalizes it to functions defined on certain subsets of the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>.

Stone–Weierstrass theorem open-in-new

Stone–Weierstrass theorem