<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 ''<math>T''</math> admits a unique [[Fixed point (mathematics)|fixed-point]] ''<math>x^*'' \in ''X''</math> (i.e. ''<math>T''(''x^*'')= ''x^*''</math>). Furthermore, ''<math>x^*''</math> can be found as follows: start with an arbitrary element ''x''<submath>0x_0\in X</submath> in ''X'' and define a [[sequence]] {''x<submath>n\{x_n\}</submath>''} by ''x<submath>x_n=T(x_{n-1})</submath>''=for ''T''(''x''<submath>''n''−1\geq1</submath>) for ''n'' ≥ 1. Then {{nowrap|''x<submath>nx_n\to x^*</submath>'' → ''x*''}}.</blockquote>
''Remark 1.'' The following inequalities are equivalent and describe the [[Rate of convergence|speed of convergence]]:
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</math>
Any such value of ''<math>q''</math> is called a ''[[Lipschitz constant]]'' for ''<math>T''</math>, and the smallest one is sometimes called "the best Lipschitz constant" of ''<math>T''</math>.
''Remark 2.'' ''In general, <math>d''(''T''(''x''), ''T''(''y'')) < '' d''(''x'', ''y'')</math> for all ''<math>x'' ≠\neq ''y''</math>, is in general not enough to ensure the existence of a fixed point, as is shown by the map ''<math>T'' :\colon [1, ∞\infty) →\to [1, ∞\infty),</math> ''by <math>T''(''x'') = ''x'' + 1/''x''</math>, which lacks a fixed point. However, if ''<math>X''</math> is [[Compact space|compact]], then this weaker assumption does imply the existence and uniqueness of a fixed point, that can be easily found as a minimizer of ''<math>d''(''x'', ''T''(''x'')),</math>. indeedIndeed, a minimizer exists by compactness, and has to be a fixed point of ''<math>T''</math>. It then easily follows that the fixed point is the limit of any sequence of iterations of ''<math>T''</math>.
''Remark 3.'' When using the theorem in practice, the most difficult part is typically to define ''<math>X''</math> properly so that ''<math>T''(''X'')\subseteq ⊆ ''X''</math>.
