|WikiProject Mathematics||(Rated B-class, Mid-importance)|
How about complex case?
Stone-Weierstrass doesn't work in the complex case. Consider complex functions on the unit circle in the complex plane. Polynomials form an algebra with a unit and containing functions which separate the points on the circle. However, the function z->complex_conjugate(z) is *not* well approximated by any polynomial. Indeed, conventional "inner product" between this function and any polynomial is 0 (because the complex conjugate of the complex conjugate is again z).
The solution is to consider *-algebras, that is, algebras which are also closed under the operation of complex conjugation. -- Miguel
Forgive the silly question, but I want to make sure I have this clear: the theorem (in the simple form) only applies to approximating functions from . It does not apply to , correct? Am I correct in saying that in general, it applies to real-valued functions where I is any compact set in the reals (i.e. any closed set)? And that it fails if I is not compact? Lavaka 17:23, 17 August 2006 (UTC)
- The general formulation (see article) starts with
- Suppose K is a compact Hausdorff space
- so compactness of the domain looks like a necessary condition to me. Perhaps we should include a Proof? --CompuChip 09:45, 31 January 2007 (UTC)
Weierstrass theorem currently redirects here
Students of different disciplines or sub-disciplines are often taught about the "Weierstrass theorem," which may refer to the extreme value theorem, Stone-Weierstrass, Bolzano-Weierstrass, or who knows what. Currently Weierstrass theorem redirects here, to Stone-Weierstrass theorem. I think we should have it redirect to a disambiguation page of sorts, perhaps a page called, "Mathematical objects bearing the name of Karl Weierstrass," which would include a list of theorems, as well as a short description of the theorem so students can figure out which one is relevant. Does this sound like a good idea? I brought this up at Talk:Karl Weierstrass as well, so please feel free to discuss it there. Smmurphy(Talk) 21:22, 18 July 2007 (UTC)
About the comment in "See also"
The "See also" section has the following comment:
- However, as is shown in Rudin's Principles of Mathematical Analysis, one can easily find a polynomial P uniformly approximating ƒ by convolving ƒ with a polynomial kernel.
I find this remark a bit strange and naive. Weierstrass' original proof is by convolving with a Gausian kernel. Is it really reasonable to mention explicitely Rudin here? I think not, and will change it, if nobody objects. --Bdmy (talk) 12:08, 3 April 2009 (UTC)
About the condition "Hausdorff"
The article says:
- f|S ∈ AS for every maximal set S ⊂ X such that AS contains no non-constant real functions.