Brouwer fixed-point theorem: Difference between revisions

Revision as of 14:02, 12 August 2003

Revision as of 14:03, 12 August 2003

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The theorem has several "real world" illustrations. Take for instance two equal size sheets of graph paper with coordinate systems on them, lay one flat on the table and crumple up (but don't rip) the other one and place it any way you like on top of the first. Then there will be at least one point of the crumpled sheet that lies exactly on top of the corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of the ''n'' = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet right beneath it.

The Brouwer Fixedfixed Pointpoint Theoremtheorem was one of the early achievements of [[algebraic topology]], and is the basis of more general fixed point theorems which are important in [[functional analysis]]. The case ''n'' = 3 was proved by [[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]] in 1909. [[Jacques Hadamard]] proved the general case in 1910, and Brouwer found a different proof in 1912.

== Proof outline ==

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== Generalizations ==

Generalizations of the Brouwer Fixedfixed Pointpoint Theoremtheorem to infinite dimensions include the [[Schauder fixed point theorem]] (if ''C'' is a [[nonempty]] [[closed set|closed]] [[convex]] subset of a [[Banach space]] and ''f'' is a continuous map from ''C'' to ''C'' whose image is [[compact|countably compact]], then ''f'' has a fixed point) and the [[Tychonoff fixed point theorem]] (if ''C'' is a nonempty compact convex subset of a [[locally convex]] [[topological vector space]], then any continuous map ''f'' from ''C'' to ''C'' has a fixed point).