Also known as the contraction mapping theorem, is a fundamental result in the theory of metric spaces, including normed spaces. It provides conditions under which a self-map of a metric space has a unique fixed point.
Theorem
Let $(X,d)$ be a complete metric space and let $f : X \to X$ be a contraction, i.e., there exists a constant $0 \leq k < 1$ such that for all $x, y \in X$, we have $d(f(x), f(y)) \leq k d(x,y)$. Then $f$ has a unique fixed point, i.e., there exists a unique $x \in X$ such that $f(x) = x$.
$\blacksquare$
Intuitively, the contraction mapping theorem says that if a map contracts distances in the space, then it has a unique fixed point.
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Author of the notes: Antonio J. Pan-Collantes
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