Talk:Stone–Weierstrass theorem

Domain question

When we say something about "pure states"... is the reader supposed to know what a pure state is??? — Preceding unsigned comment added by 189.61.2.44 (talk) 01:37, 1 November 2019 (UTC)

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

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)

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"

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)

I believe you are correct, the proof I know doesn't seem to use the Hausdorff assumption at all. AJ LaMotta (talk) 18:51, 29 May 2024 (UTC)

bishop theorem

The article says:

• f|_S ∈ A_S for every maximal set S ⊂ X such that A_S contains no non-constant real functions.

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

89.240.118.214 (talk) 15:44, 6 March 2021 (UTC)

Functions Domain: $L^{2}$ functions?

In this article we consider $C(X)$ as super function spaces. But in this MSE question, $L^{2}(X)$ is also considered when using Stone-Weierstrass theorem. Is it right? If yes, maybe we could replenish this. -- Namasikanam (talk) 03:15, 17 September 2019 (UTC)

"The Generalized Weierstrass Approximation Theorem" wrong PDF

The link following the last but one reference points to the PDF of a paper by Hager and is unrelated to the cited paper by Stone. Bill

Thanks for noticing. I removed the erroneous link. Saung Tadashi (talk) 17:02, 6 March 2021 (UTC)

Editorial history

This new section refers to an English paper published in 1885 under the title On the possibility of giving an analytic representation to an arbitrary function of real variable. The related digital copy seems not to be available in the Internet Archive. Hence, it has been temporarily sourced by four academic papers, which, in my modest opinion, can be removed when the related primary source will be eventually made available on the web.Theologian81sp (talk) 21:51, 3 July 2021 (UTC)