Editing Stone–Weierstrass theorem (section)

{{Short description|Mathematical theorem in the study of analysis}}
In [[mathematical analysis]], the '''Weierstrass approximation theorem''' states that every [[continuous function]] defined on a closed [[interval (mathematics)|interval]] {{math|[''a'', ''b'']}} can be [[uniform convergence|uniformly approximated]] as closely as desired by a [[polynomial]] 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 [[polynomial interpolation]]. The original version of this result was established by [[Karl Weierstrass]] in [[#Historical works|1885]] using the [[Weierstrass transform]].

[[Marshall H. Stone]] considerably generalized the theorem<ref>{{citation|first=M. H.|last=Stone|year=1937|author-link=Marshall Stone|title=Applications of the Theory of Boolean Rings to General Topology|journal=Transactions of the American Mathematical Society| volume=41| issue=3| pages=375–481| doi=10.2307/1989788| jstor=1989788| doi-access=free}}</ref> and simplified the proof.<ref>{{citation|doi=10.2307/3029750|first=M. H.|last=Stone|year=1948|author-link=Marshall Stone| title=The Generalized Weierstrass Approximation Theorem|journal=Mathematics Magazine| volume=21| issue=4| pages=167–184| jstor=3029750| mr=27121 }}; '''21''' (5), 237–254.</ref>  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 {{math|[''a'', ''b'']}}, an arbitrary [[compact space|compact]] [[Hausdorff space]] {{mvar|X}} is considered, and instead of the [[Algebra over a field|algebra]] of polynomial functions, a variety of other families of continuous functions on <math>X</math> are shown to suffice, as is [[Stone–Weierstrass theorem#Stone–Weierstrass theorem, real version|detailed below]].  The Stone–Weierstrass theorem is a vital result in the study of the algebra of [[continuous functions on a compact Hausdorff space]].

Further, there is a generalization of the Stone–Weierstrass theorem to noncompact [[Tychonoff space]]s, namely, any continuous function on a Tychonoff space is approximated [[compact-open topology|uniformly on compact sets]] by algebras of the type appearing in the Stone–Weierstrass theorem and described below.

A different generalization of Weierstrass' original theorem is [[Mergelyan's theorem]], which generalizes it to functions defined on certain subsets of the [[complex plane]].