Every continuous real-valued function on a compact space can be uniformly approximated as closely as desired by functions from a subalgebra that separates points and contains the constants. It generalizes the Weierstrass approximation theorem to a wide class of spaces and function algebras.
Facts
StatementThe Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on X are shown to suffice. 1 Classification
Statement Form Connections
Sources
1. Stone-Weierstrass Theorem (Wikipedia)
Wikimedia Foundationlead section, generalization sentenceQuote, lead section, generalization sentence
The Stone-Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval [a, b], an arbitrary compact Hausdorff space X is considered, and instead of the algebra of polynomial functions, a variety of other families of continuous functions on X are shown to suffice, as is detailed below.
2. Marshall H. Stone (Wikipedia)
Wikimedia Foundationbody text, 1937 publication sentenceQuote, body text, 1937 publication sentence
In 1937, he published the Stone-Weierstrass theorem which generalized Weierstrass's theorem on the uniform approximation of continuous functions by polynomials.
Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.