Talk:Brouwer fixed-point theorem: Difference between revisions

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== Did Brouwer offer the first proof for continuous functions? ==

== Accessible proof==

- In the book ''History of Topology'' by James, on pages 273-274, it says: "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."

Courant and Robbins provide an accessible proof.

<small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:198.144.199.xxx|198.144.199.xxx]] ([[User talk:198.144.199.xxx|talk]] • [[Special:Contributions/198.144.199.xxx|contribs]]) 30 August 2001</span></small><!-- Template:Unsigned -->

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.'"

== First proved by Bohl or Brouwer? ==

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?'''

According to Lyusternik ''Convex Figures and Polyhedra'', the theorem was first proved by a Lettish mathematician named Bohl. No references are provided. Anyone know what this is about?--[[User:192.35.35.36|192.35.35.36]] 00:08, 18 Feb 2005 (UTC)

== Accessible proof==

- In the book ''History of Topology'' by James, on pages 273-274, it says: "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."

Courant and Robbins provide an accessible proof.

<small><span class="autosigned">—Preceding [[Wikipedia:Signatures|unsigned]] comment added by [[User:198.144.199.xxx|198.144.199.xxx]] ([[User talk:198.144.199.xxx|talk]] • [[Special:Contributions/198.144.199.xxx|contribs]]) 30 August 2001</span></small><!-- Template:Unsigned -->

== Citation style ==