Brouwer's Fixed Point Theorem
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Theorem
One-Dimensional Version
Let $f: \closedint a b \to \closedint a b$ be a real function which is continuous on the closed interval $\closedint a b$.
Then:
- $\exists \xi \in \closedint a b: \map f \xi = \xi$
That is, a continuous real function from a closed real interval to itself fixes some point of that interval.
Smooth Mapping
A smooth mapping $f$ of the closed unit ball $B^n \subset \R^n$ into itself has a fixed point:
- $\forall f \in \map {C^\infty} {B^n \to B^n}: \exists x \in B^n: \map f x = x$
General Case
A continuous mapping $f$ of the closed unit ball ${B^n}^- \subset \R^n$ into itself has a fixed point:
- $\forall f \in \map {C^0} { {B^n}^- \to {B^n}^-} : \exists x \in {B^n}^- : \map f x = x$
Also known as
Brouwer's Fixed Point Theorem is also known just as Brouwer's Theorem.
Also see
Source of Name
This entry was named for Luitzen Egbertus Jan Brouwer.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Brouwer's theorem
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Brouwer's theorem