Banach Fixed Point Theorem

Banach Fixed Point Theorem (contraction mapping principle): Let $(X,d)$ be a complete metric space and let $T:X\to X$ be a contraction with contraction constant $c\in[0,1)$. Then:

• There exists a unique fixed point $x^\ast\in X$ such that $T(x^\ast)=x^\ast$.
• For any starting point $x_0\in X$, the iterates defined by $x_{n+1}=T(x_n)\quad(n\ge 0)$ converge to $x^\ast$.
• Quantitative error bounds hold: for all $n\ge 0$, $d(x_{n},x^\ast)\le \frac{c^{n}}{1-c}\,d(x_1,x_0), \qquad d(x_n,x^\ast)\le c^n\,d(x_0,x^\ast).$

This theorem is one of the main uses of completeness: it turns a global “shrinking” hypothesis into existence and uniqueness of solutions of $T(x)=x$.