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Difference between revisions of "Stone-Weierstrass theorem"

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A broad generalization of the classical [[Weierstrass theorem|Weierstrass theorem]] on the approximation of functions, due to M.H. Stone (1937). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s0903701.png" /> be the ring of continuous functions on a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s0903702.png" /> with the topology of uniform convergence, i.e. the topology generated by the norm
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A broad generalization of the classical [[Weierstrass theorem|Weierstrass theorem]] on the approximation of functions, due to M.H. Stone (1937). Let $C(X)$ be the ring of continuous functions on a compactum $X$ with the topology of uniform convergence, i.e. the topology generated by the norm
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s0903703.png" /></td> </tr></table>
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$$\|f\| = \max_{x\in X}\left|f(x)\right|, \quad f\in C(X),$$
  
and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s0903704.png" /> be a subring containing all constants and separating the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s0903705.png" />, i.e. for any two different points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s0903706.png" /> there exists a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s0903707.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s0903708.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s0903709.png" />, i.e. every continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s09037010.png" /> is the limit of a uniformly converging sequence of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090370/s09037011.png" />.
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and let $C_0\subseteq C(X)$ be a subring containing all constants and separating the points of $X$, i.e. for any two different points $x_1, x_2\in X$ there exists a function $f\in C_0$ for which $f(x_1)\neq f(x_2)$. Then $[C_0]=C(X)$, i.e. every continuous function on $X$ is the limit of a uniformly converging sequence of functions in $C_0$.
  
  

Latest revision as of 13:13, 12 December 2013

A broad generalization of the classical Weierstrass theorem on the approximation of functions, due to M.H. Stone (1937). Let $C(X)$ be the ring of continuous functions on a compactum $X$ with the topology of uniform convergence, i.e. the topology generated by the norm

$$\|f\| = \max_{x\in X}\left|f(x)\right|, \quad f\in C(X),$$

and let $C_0\subseteq C(X)$ be a subring containing all constants and separating the points of $X$, i.e. for any two different points $x_1, x_2\in X$ there exists a function $f\in C_0$ for which $f(x_1)\neq f(x_2)$. Then $[C_0]=C(X)$, i.e. every continuous function on $X$ is the limit of a uniformly converging sequence of functions in $C_0$.


Comments

The expository article [a4] is recommended in particular.

References

[a1] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
[a2] A. Schönhage, "Approximationstheorie" , de Gruyter (1971)
[a3] M.H. Stone, "The generalized Weierstrass approximation theorem" Math. Mag. , 21 (1948) pp. 167–183; 237–254
[a4] M.H. Stone, "A generalized Weierstrass approximation theorem" R.C. Buck (ed.) , Studies in Modern Analysis , 1 , Math. Assoc. Amer. (1962) pp. 30–87
[a5] M.H. Stone, "Applications of the theory of Boolean rings to general topology" Trans. Amer. Math. Soc. , 41 (1937) pp. 375–481
How to Cite This Entry:
Stone-Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stone-Weierstrass_theorem&oldid=17999
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article