Several converses of the Banach contraction principle exist. The following is due to [[Czesław Bessaga]], from 1959:
Let ''<math>f''\,\colon : ''X''\to → ''X''</math> be a map of an abstract [[set (mathematics)|set]] such that each [[iterated function|iterate]] ''f<supmath>f^n</supmath>'' has a unique fixed point. Let ''<math>q'' ∈ \in(0,1)</math>, then there exists a complete metric on ''<math>X''</math> such that ''<math>f''</math> is contractive, and ''<math>q''</math> is the contraction constant.
Indeed, very weak assumptions suffice to obtain such a kind of converse. For example if ''<math>f''\,\colon : ''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]] ''<math>a''</math>, such that for each ''<math>x'' \in ''X''</math> we have ''f<supmath>f^n</sup>''(''x'')\mapsto → ''a''</math>, then there already exists a metric on ''<math>X''</math> with respect to which ''<math>f''</math> satisfies the conditions of the Banach contraction principle with contraction constant <math>1/2</math>.<ref>{{cite journal |first=Pascal |last=Hitzler | authorlink1=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]].
