Contraction map

Contraction map theorem

The contraction map theorem applies to contracting endomorphisms of complete metric spaces in .

Let be a non-empty Complete metric space and be a Contraction map. Then has a unique fixed point , i.e. such that . #m/thm/anal

Proof (sketch)

The uniqueness part of the theorem is easy to prove, for if there exist such that and , then meaning the distance was not contracted.

The existence part is proven using a sequence of repeated applications of , which must be a Cauchy sequence since distances contract upon each subsequent application. The limit of this sequence1 is .

#state/tidy | #lang/en | #SemBr

Footnotes

  1. Which exists by completeness. ↩