Editing Banach fixed-point theorem (section)

==Converses==
Several converses of the Banach contraction principle exist. The following is due to [[Czesław Bessaga]], from 1959:

Let ''f'' : ''X'' → ''X'' be a map of an abstract [[set (mathematics)|set]] such that each [[iterated function|iterate]] ''f<sup>n</sup>'' has a unique fixed point. Let <math>q \in (0, 1),</math> then there exists a complete metric on ''X'' such that ''f'' is contractive, and ''q'' is the contraction constant.

Indeed, very weak assumptions suffice to obtain such a kind of converse. For example if <math>f : X \to X</math> is a map on a [[T1 space|''T''<sub>1</sub> topological space]] with a unique [[fixed point (mathematics)|fixed point]] ''a'', such that for each <math>x \in X</math> we have ''f<sup>n</sup>''(''x'') → ''a'', then there already exists a metric on ''X'' with respect to which ''f'' satisfies the conditions of the Banach contraction principle with contraction constant 1/2.<ref>{{cite journal |first1=Pascal |last1=Hitzler | author-link1=Pascal Hitzler|first2=Anthony K. |last2=Seda |title=A 'Converse' of the Banach Contraction Mapping Theorem |journal=Journal of Electrical Engineering |volume=52 |issue=10/s |year=2001 |pages=3–6 }}</ref> In this case the metric is in fact an [[ultrametric]].