Contraction map theorem

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

Let $(𝑋, 𝑑)$ be a non-empty Complete metric space and $𝑇 : 𝑋 \to 𝑋$ be a Contraction map. Then $𝑇$ has a unique fixed point $𝑥 0 \in 𝑋$ , i.e. such that $𝑇 (𝑥 0) = 𝑥 0$ . #m/thm/anal

Proof (sketch)

The uniqueness part of the theorem is easy to prove, for if there exist $𝑥 \neq 𝑦$ such that $𝑇 (𝑥) = 𝑥$ and $𝑇 (𝑦) = 𝑦$ , then $𝑑 (𝑇 (𝑥), 𝑇 (𝑦)) = 𝑑 (𝑥, 𝑦)$ meaning the distance $𝑑 (𝑥, 𝑦)$ was not contracted.

The existence part is proven using a sequence of repeated applications of $𝑇$ , which must be a Cauchy sequence since distances contract upon each subsequent application. The limit of this sequence¹ is $𝑥 0$ .

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Which exists by completeness. ↩

Contraction map theorem

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Contraction map theorem

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