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 | + | 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 |
| − | + | $$\|f\| = \max_{x\in X}\left|f(x)\right|, \quad f\in C(X),$$ | |
| − | and let | + | 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 $C_0$. |
Revision as of 10:02, 14 February 2012
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=21026
Stone-Weierstrass theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stone-Weierstrass_theorem&oldid=21026
This article was adapted from an original article by V.I. Ponomarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article