Banach fixed-point theorem: Difference between revisions

Revision as of 22:33, 30 August 2023

Revision as of 04:24, 13 October 2023

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==Statement==

''Definition.'' Let <math>(X, d)</math> be a [[complete metric space]]. Then a map <math>T : X \to X</math> is called a [[contraction mapping]] on ''X'' 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 \in X.</math>