PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (232)
Orphanage
Unclass'd
Unproven (519)
Corrections (23)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Very high Entry average rating: No information on entry rating
Brouwer fixed point theorem (Theorem)

Theorem Let $ \textbf{B}=\{x\in\mathbb{R}^n: \left\Vert x\right\Vert\le 1\}$ be the closed unit ball in $ \mathbb{R}^n$. Any continuous function $ f: \textbf{B}\to\textbf{B}$ has a fixed point.

Notes

Shape is not important
The theorem also applies to anything homeomorphic to a closed disk, of course. In particular, we can replace $ \textbf{B}$ in the formulation with a square or a triangle.
Compactness counts (a)
The theorem is not true if we drop a point from the interior of $ \textbf{B}$. For example, the map $ f(\vec{x})=\frac{1}{2}\vec{x}$ has the single fixed point at 0; dropping it from the domain yields a map with no fixed points.
Compactness counts (b)
The theorem is not true for an open disk. For instance, the map $ f(\vec{x})=\frac{1}{2}\vec{x}+(\frac{1}{2},0,\ldots,0)$ has its single fixed point on the boundary of $ \textbf{B}$.



"Brouwer fixed point theorem" is owned by mathcam. [ full author list (3) | owner history (2) ]
(view preamble | get metadata)

View style:

See Also: fixed point, Schauder fixed point theorem, Tychonoff fixed point theorem, KKM lemma, KKM lemma

Keywords:  fixed point, nonconstructive

Attachments:
proof of Brouwer fixed point theorem (Proof) by bwebste
Brouwer fixed point in one dimension (Proof) by mathcam
proof of Brouwer fixed point theorem (Proof) by uriw
Log in to rate this entry.
(view current ratings)

Cross-references: boundary, open, domain, map, interior, point, triangle, square, homeomorphic, fixed point, continuous function, unit ball, closed
There are 6 references to this entry.

This is version 4 of Brouwer fixed point theorem, born on 2002-06-05, modified 2007-06-24.
Object id is 3046, canonical name is BrouwerFixedPointTheorem.
Accessed 12286 times total.

Classification:
AMS MSC: 47H10 (Operator theory :: Nonlinear operators and their properties :: Fixed-point theorems)
 54H25 (General topology :: Connections with other structures, applications :: Fixed-point and coincidence theorems)
 55M20 (Algebraic topology :: Classical topics :: Fixed points and coincidences)
Pending Errata and Addenda
None.
[ View all 3 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | prove | add result | add corollary | add example | add (any)