''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 <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 <math>x_0\in X</math> and define a [[sequence]] <math>\{x_n\}</math> by <math>x_n=T(x_{n-1})</math> for <math>n\geq1</math>. Then <math>x_n\to x^*</math>.</blockquote>
''Remark 1.'' The following inequalities are equivalent and describe the [[Rate of convergence|speed of convergence]]:
