= Contraction mapping theorem
{wiki=Banach_fixed-point_theorem}
= Banach fixed-point theorem
{c}
{synonym}
A <contraction mapping> $T:X\to X$ on a nonempty <complete metric space> has exactly one <fixed point> $x_*$, and the <sequence> $x_{n+1}=T(x_n)$ converges to $x_*$ from every $x_0\in X$. The same constant $0\leq L<1$ must satisfy $d(Tx,Ty)\leq Ld(x,y)$ for all $x,y\in X$. The statement applies to <metric spaces>, including closed subsets of a <Banach space>; neither linearity nor a <vector space> structure is required.https://maths.dur.ac.uk/users/norbert.peyerimhoff/analysis0708/dirknotes.pdf[Dirk Schütz, Analysis III/IV, page 13: contraction mapping theorem]{ref}
The force of the result is that existence, uniqueness, and a practical approximation procedure arrive together. Repeated application produces corrections bounded by a <geometric series>, hence a <Cauchy sequence>; <completeness> supplies its limit. The <Lipschitz continuity> of $T$ makes that limit a <fixed point>. Two different <fixed points> would have their positive distance multiplied by at most $L<1$ while remaining unchanged, which is impossible.
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