Brouwer's Fixed Point Theorem/General Case

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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




Sources