Brouwer's Fixed Point Theorem/General Case

Theorem

A continuous mapping $f$ of the closed unit ball $\overline B^n \subset \R^n$ into itself has a fixed point:

$\forall f \in \map {C^0} {\overline B^n \to \overline B^n} : \exists x \in \overline B^n : \map f x = x$

Proof

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Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): fixed-point theorem
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): fixed-point theorem

• Weisstein, Eric W. "Brouwer Fixed Point Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrouwerFixedPointTheorem.html