Brouwer fixed point theoremBrouwerFixedPointTheorem2002-06-05 08:47:112007-06-24 16:15:12Theoremfixed pointnonconstructive% this is the default PlanetMath preamble. as your knowledge
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\newcommand{\Complex}{\mathbb{C}}{\bf Theorem}
Let $\textbf{B}=\{x\in\Reals^n: \norm{x}\le 1\}$ be the closed unit ball in
$\Reals^n$. Any continuous function $f: \textbf{B}\to\textbf{B}$ has a fixed point.
\subsection*{Notes}
\begin{description}
\item[Shape is not important]
The theorem also applies to anything homeomorphic to a closed disk, of course. In particular, we can replace $\textbf{B}$ in the formulation with a square or
a triangle.
\item[Compactness counts (a)]
The theorem is not true if we drop a point from the interior of $\textbf{B}$. 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 \PMlinkname{fixed points}{FixedPoint}.
\item[Compactness counts (b)]
The theorem is not true for an open disk. For instance, the map $f(\vec{x})=\frac{1}{2}\vec{x}+(\frac{1}{2},0,\ldots,0)$ has its single fixed point on the boundary of $\textbf{B}$.
\end{description}