Brouwer's Fixed Point Theorem

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Brouwer's Fixed Point Theorem

The Brouwer fixed point theorem is the statement that a self map of a convex subset of Rⁿ always has a fixed point. We shall prove this for the disk.

Theorem 6.10 Let D² denote the closed disk. Then any self map f : D² $\longrightarrow$ D² has a fixed point, namely, there is some x $\in$ D² such that f (x) = x.

$\begin{proof}Assume the theorem is false. Thus there exists a self map $f$\space... ... thus we obtain a contradiction completing the proof of the theorem. \end{proof}$

Ran Levi
2000-03-13