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Brouwer fixed point theorem
Theorem Let $\textbf{B}=\{x\in\mathbb{R}^{n}:\left\x\right\\leq 1\}$ 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 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 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)
Keywords:
fixed point, nonconstructive
Related:
FixedPoint, SchauderFixedPointTheorem, TychonoffFixedPointTheorem, KKMlemma, KKMLemma
Type of Math Object:
Theorem
Major Section:
Reference
Mathematics Subject Classification
55M20 no label found54H25 no label found47H10 no label found Forums
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new correction: Error in proof of Proposition 2 by alex2907
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new correction: Error in proof of Proposition 2 by alex2907
Jun 24
new question: A good question by Ron Castillo
Jun 23
new question: A trascendental number. by Ron Castillo
Jun 19
new question: Banach lattice valued Bochner integrals by math ias
Jun 13
new question: young tableau and young projectors by zmth
Jun 11
new question: binomial coefficients: is this a known relation? by pfb