Recall that a contraction mapping from a metric space to itself is a Lipschitz function with constant strictly less than 1.
Let us look at $\mathbb{R}^d$. I need to find a norm $\|\cdot\|$ and a linear mapping $T:\mathbb{R}^d\to\mathbb{R}^d$ such that $T$ is a contraction mapping with respect to $\|\cdot\|$ but not in the Euclidean norm. First I thought on maybe taking $T(v) = \frac{1}{\sqrt{d}}v$ with the max norm, but then I realized that from the homogeneity of the norm, I figure this mapping cannot be $T(v) = av$ for some $a<1$, because then $\|T(v) - T(w)\|_2 = a\|v-w\|_2$ ($\|\cdot\|_2$ is the Euclidean norm).
