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The '''Banach [[fixed point theorem]]''' is an important tool in the theory of [[metric space]]s; it guarantees the existence and uniqueness of [[fixed point (mathematics)|fixed points]] of certain self maps of metric spaces, and provides a constructive method to find those fixed points. The theorem is named after [[Stefan Banach]] (1892-1945), and was first stated by Banach in [[1922]].
== The theorem==
Let (''X'', d) be a non-empty [[complete metric space]]. Let ''T'' : ''X'' <tt>-></tt> ''X'' be a ''[[contraction mapping]]'' on ''X'', i.e: there is a [[real number]] ''q'' < 1 such that
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