  We study necessary and sufficient conditions for contraction and incremental
stability of dynamical systems with respect to non-Euclidean norms. First, we
introduce weak pairings as a framework to study contractivity with respect to
arbitrary norms, and characterize their properties. We introduce and study the
sign and max pairings for the $\ell_1$ and $\ell_\infty$ norms, respectively.
Using weak pairings, we establish five equivalent characterizations for
contraction, including the one-sided Lipschitz condition for the vector field
as well as matrix measure and Demidovich conditions for the corresponding
Jacobian. Third, we extend our contraction framework in two directions: we
prove equivalences for contraction of continuous vector fields and we formalize
the weaker notion of equilibrium contraction, which ensures exponential
convergence to an equilibrium. Finally, as an application, we provide (i)
incremental input-to-state stability and finite input-state gain properties for
contracting systems, and (ii) a general theorem about the Lipschitz
interconnection of contracting systems, whereby the Hurwitzness of a gain
matrix implies the contractivity of the interconnected system.
