From: ullrich@math.okstate.edu (David C. Ullrich)
Subject: Re: Need wording of STONE-Weierstrass Theorem
Date: Mon, 30 Oct 2000 15:23:58 GMT
Newsgroups: sci.math.num-analysis,sci.math
Summary: [missing]

On Mon, 30 Oct 2000 14:16:32 +0200, "Ihor Smal" <ihor@city.lviv.net>
wrote:

>Hi All
>
>Could anybody write me the wording of the of Stone-Weierstrass Theorem?

	Comes in various versions. Say K is a compact Hausdorff space.
C(K) will be the space of all complex-valued continuous functions on 
K. A _subalgebra_ of C(K) is a linear subspace which is closed under
multiplication. A subalgebra A is "self-adjoint" if it is closed under
taking complex conjugates (if f is in A and if g(x) is the complex
conjugate of f(x) for all x in K then g is in A.) A "separates points"
if fx, y in K, x <> y implies that there exists f in A with 
f(x) <> f(y).

	The theorem says that a self-adjoint subalgebra of C(K)
that separates points and contains the constant functions must 
be dense in C(K) (that's "dense" with respect to the uniform 
metric d(f,g) = sup |f(x) - g(x)|.)

>Or maybe someone knows if that theorem allows to use negative values of
>degrees for the basic approximating functions (n-dim. polynoms) when one
>would like to approximate some given function.

	Whether you can do that would depend on exactly
what the situation was. Approximating functions on what set?

>Thank you in advance,
>
>Ihor Smal