L0 sparsity count ($\Vert x{\Vert }_0=|\mathrm{supp}x|$)

The number of nonzero coordinates of a finite vector is its L0 sparsity count. It vanishes only at zero, but is unchanged under multiplication by a nonzero scalar, so it is not a norm. Minimizing it under linear measurement constraints finds a sparse vector with the smallest possible support of a vector. Sparse injectivity of order $s$ guarantees unique recovery of every sparse vector with at most $s$ nonzero coordinates by this objective.