= Schauder fixed-point theorem
{c}
{title2=$T(K)\subset K\Longrightarrow\exists u\in K:T(u)=u$}
Every <continuous map> from a nonempty compact <convex set> $K$ in a <Banach space> to itself has a <fixed point>. A useful equivalent form takes a nonempty closed bounded <convex set> $C$ and a <continuous map> $T:C\to C$ whose image is <relatively compact>. To obtain this version from the compact-domain version, restrict $T$ to $\overline{\operatorname{conv}}T(C)\subset C$ and use <closed convex hull of a compact set>.
For a proof on $K$, the <Schauder approximation lemma> gives <continuous maps> $T_\varepsilon:K\to K\cap L_\varepsilon$ with finite-dimensional $L_\varepsilon$ and $\|T-T_\varepsilon\|_\infty\leq\varepsilon$. The <Brouwer fixed-point theorem> on the nonempty compact <convex set> $K\cap L_\varepsilon$ gives $x_\varepsilon=T_\varepsilon x_\varepsilon$, so $\|Tx_\varepsilon-x_\varepsilon\|\leq\varepsilon$. A <convergent subsequence> as $\varepsilon\downarrow0$ and the <continuity> of $T$ give an exact <fixed point>.
Back to article page
