Talk:Stone–Weierstrass theorem

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 Field: Analysis

How about complex case?[edit]

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[edit]

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[edit]

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"[edit]

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"[edit]

Is the condition "Hausdorff" necessary? It seems that "compact" is enough. --Hang —Preceding unsigned comment added by 219.236.149.32 (talk) 12:55, 28 April 2011 (UTC)

bishop theorem[edit]

The article says:

  • f|SAS for every maximal set SX such that AS contains no non-constant real functions.

So what is AS? --11:22, 30 March 2012 (UTC) — Preceding unsigned comment added by 141.35.13.182 (talk)