Banach fixed-point theorem

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Educational level: this is a tertiary (university) resource.

Banach fixed-point theorem. Let $(X,d)$ be a non-empty complete metric space i.e. the space with a metric where every Cauchy sequence i.e. such that its elements are all arbitrarily closely bunched in the sense of the metric if their indexes are sufficiently high is convergent with a contraction mapping $T:X\to X.$ i.e if there exists $q\in [0,1)$ such that

$d(T(x),T(y))\leq qd(x,y)$

i.e. that the metric between the images of any of two elements is contracted or scaled by the factor less than 1 with respect to the metric between elements themself for all $x,y\in X$. Then there is only one unique fixed point fixed-point for $T$ $x^{*}$ in $X$ (i.e. $T(x^{*})=x^{*}$ or the element such that it image is the same that the element). Furthermore, $x^{*}$ can be found by the infinite nesting of T or the infinite superposition $T((((...x))$ applied on any element of the space $x\in X$ has the fixed point limit: start with an arbitrary element $x_{0}\in X$ and define a sequence $(x_{n})_{n\in \mathbb {N} }$ by $x_{n}=T(x_{n-1})$ for $n\geq 1.$ Then $\lim _{n\to \infty }x_{n}=x^{*}$.

Example of numerical applications - calculating high accuracy π

Banach theorem allows for example fast and accurate calculation of the π number using the trigonometric functions which numerically are the power Taylor series.

Because $\sin(\pi )=0$ and the π is the fixed point of for example the function $f(x)=\sin(x)+x$

i.e.

$f(\pi )=\pi$

and also the function $f$ is around π the contraction mapping from the obvious reasons because its derivative at π vanishes at the bending point therefore π can be obtained from the infinite superposition for example for the argument value 3:

$\pi =f(f(f(...f(3)...))))$

Already the triple superposition of this function at $3$ gives π with accuracy to 33 digits:

$f(f(f(3)))=3.141592653589793238462643383279502...$ .