Contraction map theorem
The contraction map theorem applies to contracting endomorphisms of complete metric spaces in Category of topological spaces.
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 .
Footnotes
-
Which exists by completeness. ↩