PlanetMath: Stone-Weierstrass theorem

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Stone-Weierstrass theorem

(Theorem)

Let be a compact space and let $C^0(X,\mathbb{R})$ be the algebra of continuous real functions defined over . Let $\mathcal{A}$ be a subalgebra of $C^0(X,\mathbb{R})$ for which the following conditions hold:

$\forall x, y \in X, x \ne y, \exists f \in \mathcal{A} : f(x) \neq f(y)$
$1 \in \mathcal{A}$

Then $\mathcal{A}$ is dense in $C^0(X,\mathbb{R})$ .

This theorem is a generalization of the classical Weierstrass approximation theorem to general spaces.

"Stone-Weierstrass theorem" is owned by rspuzio. [ full author list (2) | owner history (1) ]

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Attachments:: proof of Stone-Weierstrass theorem (Proof) by rspuzio
Stone-Weierstrass theorem (complex version) (Theorem) by asteroid

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Cross-references: Weierstrass approximation theorem, dense in, subalgebra, real functions, continuous, algebra, compact
There are 3 references to this entry.

This is version 6 of Stone-Weierstrass theorem, born on 2002-06-01, modified 2008-05-04.
Object id is 2984, canonical name is StoneWeierstrassTheorem.
Accessed 5250 times total.

Classification:

AMS MSC:

46E15 (Functional analysis :: Linear function spaces and their duals :: Banach spaces of continuous, differentiable or analytic functions)

Pending Errata and Addenda

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