I understood a version of contraction mapping theorem in a set-up of metric space, $(S,d)$.
Can someone provide an example of a linear transform in a finite-dimensional vector space that is also a contraction mapping (e.g. an operator brings two elements of the domain strictly closer in terms of $\textit{distance}$ than by themselves? When studying linear algebra in finite-dimensional vector space, is there such an example?
Best Answer
What about $$\mathbb{R} \to \mathbb{R},\quad x \mapsto \frac{1}{2}\cdot x.$$