Talk:Brouwer fixed-point theorem
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Courant and Robbins provide an accessible proof. —Preceding unsigned comment added by 198.144.199.xxx (talk • contribs) 30 August 2001
This article mixes parenthetical referencing with footnoted references. The parenthetical ones were there first, so according to WP:CITEVAR we'd have to use that until explicit consensus. However, it would be significantly easier to turn the couple of parenthetical ones into footnotes than about 50 footnotes into parentheticals. Can we form consensus to continued using footnoted references? – Finnusertop (talk ⋅ contribs) 19:59, 24 February 2019 (UTC)
- I'm sure that'd be okay here. –Deacon Vorbis (carbon • videos) 00:04, 25 February 2019 (UTC)
- Great. I've turned the remaining parentheticals into footnotes. – Finnusertop (talk ⋅ contribs) 00:10, 25 February 2019 (UTC)
Function mapping in closedness section
It is stated that the function f(x) = (x+1)/2 is a continous function from the open interval (-1,1) to itself. Is it not the case that the function maps from (-1,1) to (0,1)? Salomonaber (talk) 00:13, 11 March 2020 (UTC)
- It doesn't claim (nor is it required) that the function is surjective, so what's there is correct and appropriate. The example could have even arranged for a bijection, but I don't think it matters much either way. –Deacon Vorbis (carbon • videos) 00:32, 11 March 2020 (UTC)
Highly skeptical that the remarks "said to have [been] added" by Brouwer are actually due to him
Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet."
The citation is apparently from a French-language educational TV show (https://archive.is/20130113210953/http://archives.arte.tv/hebdo/archimed/19990921/ftext/sujet5.html). The remarks appear to be spoken by a fictional Brouwer trying to explain his result. The web page that this refers to gives no citation.
I would like to know who originally came up with the "crumpled paper theorem" explanation of the BFPT. It could have been Brouwer himself, but my guess is it was not. — Preceding unsigned comment added by Natkuhn (talk • contribs) 01:06, 8 May 2020 (UTC)
- Oops, yeah, that's a good catch. This probably deserves some looking into. –Deacon Vorbis (carbon • videos) 01:19, 8 May 2020 (UTC)
Did Brouwer offer the first proof for continuous functions?
In the book History of Topology by James on pages 273-274: "Bohl's theorem is also equivalent to the Brouwer theorem. Bohl's theorem was published in 1904, with a proof that required that f be differentiable. Brouwer published his fixed point theorem, for continuous functions on the 3-ball, in 1909. When the first proof for the n-ball, with f differentiable, appeared in print a year later, in an appendix by J. Hadamard to a text by Tannery, the theorem was called the 'Brouwer Fixed Point Theorem', which suggests that the result was already famous by that time. It is not known in what year Brouwer made his discovery and, apparently, communicated it to other mathematicians in an informal manner. The first published proof of the general case, that is, for continuous functions on the n-ball, was by Brouwer himself in 1912."
In the book Brouwer Degree. The Core of Nonlinear Analysis by Mawhin on page 393: "In 1910, Jules Tannery published the second volume of the second edition of his book ‘Introduction à la théorie des fonctions de variables réelles’, for the time and still now a very modern presentation of analysis, introducing Weierstrass’ rigor in France. This volume two ended with a Note of Jacques Hadamard, connected in the following way to Tannery’s book material: 'The proof, following M. Ames, of Jordan’s theorem on closed curves without double point is based upon the concept of order of a point or, equivalently on the consideration of the variation of the argument. The generalization to the case where the dimension is larger than two is given by the Kronecker index. It is a now classical notion, mainly since the publication of the Traité d’Analyse of Mr. Picard (T. I, p. 123; T. II, p. 193). It has received new applications in various recent works. My aim is to present here some of them. All the following reasonings [...] only use the continuity of the considered functions.'"
The previous quote from Mawhin's book has a strange omission ([...]) that may indicate that Hadamard did not write down a proof for the continuous case. Can someone please check out if Hadamard proved the general case for differentiable functions and not for continuous functions?
- Comments. Please sign your posts with ~~~~ and, per WP:TPG, add comments at the end of the talk page. Please stop edit-warring to insert your own point of view. There is already a discussion of the history of the FPT in the section Brouwer fixed-point theorem#First proofs. Bohl's proof applied to three dimensions. There are several historical accounts of the FPT, notably "A history of algebraic and differential topology, 1900–1960" by Jean Dieudonné. Mathsci (talk) 17:43, 21 February 2022 (UTC)