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# Banach Fixed-Point Theorem

 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z overview

If is a contraction of a non-empty closed set , i.e., if
• ,
• ,
with , then has a unique fixed point . Starting with any point , can be approximated by the sequence

The error satisfies

i.e., the iteration converges linearly.

More generally, the fixed point theorem holds in complete metric spaces. Since the proof neither requires translation invariance nor homogeneity of the norm, can be replaced by a general distance function .

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