Stone–Weierstrass theorem

E480871

approximation theorem theorem in functional analysis

The Stone–Weierstrass theorem is a fundamental result in functional analysis that characterizes when a subalgebra of continuous functions on a compact space is dense, thereby generalizing classical polynomial approximation results.

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Statements (47)

Predicate	Object
instanceOf	approximation theorem ⓘ theorem in functional analysis ⓘ
appliesTo	compact Hausdorff spaces ⓘ subalgebras of C(X) ⓘ
assumes	A contains the constant functions ⓘ A is a subalgebra of C(X) ⓘ A separates points of X ⓘ X is a compact Hausdorff space ⓘ
characterizes	density of subalgebras in C(X) ⓘ
complexVersionConclusion	A is uniformly dense in C(X,ℂ) ⓘ
complexVersionCondition	A contains the constant functions ⓘ A is closed under complex conjugation ⓘ A separates points of X ⓘ
concerns	algebras of complex-valued continuous functions ⓘ algebras of real-valued continuous functions ⓘ subalgebras closed under pointwise addition and multiplication ⓘ uniform approximation of continuous functions ⓘ
concludes	A is dense in C(X) with respect to the uniform norm ⓘ every continuous function can be uniformly approximated by elements of A ⓘ
domain	continuous functions on compact spaces ⓘ
field	approximation theory ⓘ functional analysis ⓘ topology ⓘ
generalizes	Weierstrass approximation theorem NERFINISHED ⓘ polynomial approximation on compact intervals ⓘ
hasVariant	complex Stone–Weierstrass theorem NERFINISHED ⓘ real Stone–Weierstrass theorem NERFINISHED ⓘ
historicalContext	20th-century development in functional analysis ⓘ
implies	density of polynomials in C([a,b]) ⓘ trigonometric polynomial approximation on the circle ⓘ
introducedBy	Marshall Harvey Stone NERFINISHED ⓘ
motivation	extension of polynomial approximation to general compact spaces ⓘ
namedAfter	Karl Weierstrass NERFINISHED ⓘ Marshall Harvey Stone NERFINISHED ⓘ
realVersionConclusion	A is uniformly dense in C(X,ℝ) ⓘ
realVersionCondition	A contains the constant functions ⓘ A separates points of X ⓘ
relatedTo	Banach algebras NERFINISHED ⓘ C(X) as a commutative C*-algebra ⓘ Gelfand representation theory ⓘ
topology	uniform norm topology ⓘ
typeOfDensity	uniform density ⓘ
usedIn	C*-algebra theory ⓘ harmonic analysis ⓘ potential theory ⓘ probability theory on compact spaces ⓘ spectral theory ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Weierstrass approximation theorem → generalizationOf → Stone–Weierstrass theorem ⓘ

Weierstrass approximation theorem → relatedTo → Stone–Weierstrass theorem ⓘ