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{{topospaces version at|Stone-Weierstrass theorem}}

==Statement==

Let <math>X</math> be a [[compact Hausdorff space]] and <math>C(X,\R)</math> denote the algebra of continuous functions from <math>X</math> to <math>\R</math>. Endow <math>C(X,\R)</math> with the [[topology of uniform convergence]].

Suppose <math>A</math> is a subalgebra of <math>C(X,\R)</math> such that:

* <math>A</math> is ''unital'', in the sense that <math>A</math> contains the constant function <math>1</math>
* <math>A</math> ''separates points'', in the sense that if <math>x \ne y \in X</math>, then there exists <math>f \in A</math> such that <math>f(x) \ne f(y)</math>

Then <math>A</math> is a dense subalgebra of <math>C(X,\R)</math>. In particular, if we assume <math>A</math> is closed in <math>C(X,\R)</math>, we obtain that <math>A = C(X,\R)</math>.

==Applications==

The Stone-Weierstrass theorem is used to prove that certain restricted spaces of functions, are dense in the space of all continuous functions, for some pairs of spaces.

For a complete list of applications, refer:

[[:Category:Applications of Stone-Weierstrass theorem]]

==Proof==

The result is an application of the [[Weierstrass approximation theorem]], and some clever arguments.

Stone-Weierstrass theorem - Revision history

Vipul: 1 revision

Vipul at 23:07, 13 January 2008