# Talk:Stone–Weierstrass theorem

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Field: Analysis

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

## Domain question

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 $C[a,b]\,$. It does not apply to $C(-\infty,\infty)$, correct? Am I correct in saying that in general, it applies to real-valued functions $f \in C[I]\,$ 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)

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)