''Definition.'' Let <math>(''X'',''d'')</math> be a [[complete metric space]]. Then a map <math>''T\colon'' : ''X'' \to→ ''X</math>'' is called a [[contraction mapping]] on <math>''X</math>'' if there exists <math>''q\in'' ∈ [0, 1)</math> such that
:<math>d(T(x),T(y)) \le q d(x,y)</math>
for all <math>''x'', ''y</math>'' in <math>''X</math>''.
<blockquote>'''Banach Fixed Point Theorem.''' Let <math>''(X,d)</math>'' be a [[Empty set|non-empty]] [[complete metric space]] with a contraction mapping <math>''T\colon'' : ''X'' \to→ ''X</math>''. Then ''T'' admits a unique [[Fixed point (mathematics)|fixed-point]] ''x*'' in ''X'' (i.e. ''T''(''x*'') = ''x*''). Furthermore, ''x*'' can be found as follows: start with an arbitrary element ''x''<sub>0</sub> in ''X'' and define a [[sequence]] {''x<sub>n</sub>''} by ''x<sub>n</sub>'' = ''T''(''x''<sub>''n''−1</sub>) for ''n'' ≥ 1. Then {{nowrap|''x<sub>n</sub>'' → ''x*''}}.</blockquote>
''Remark 1.'' The following inequalities are equivalent and describe the [[Rate of convergence|speed of convergence]]:
