[[Analysis MOC]]
# Contraction map
A **contraction map** is a function $f:X \to Y$ between metric spaces $(X, d_{X})$ and $(Y, d_{Y})$ where for some $0 \leq L < 1$
$$
\begin{align*}
d_{Y}(f(x), f(y)) \leq L\,d_{Y}(x,y)
\end{align*}
$$
for all $x,y \in X$. #m/def/anal 
Such a function is herefore a [[Lipschitz continuity|Lipschitz continuous]] function for which $d_{Y}(f(x), f(y)) < d(x,y)$. 

## Properties
- [[Contraction map theorem]] states contracting endomorphisms of complete metric spaces have a fixed point.


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