Norm

A function ||·|| from a vector space V over R or C to the nonnegative reals satisfying (1) ||v|| ≥ 0 with ||v|| = 0 iff v = 0 (positive definiteness), (2) ||αv|| = |α|·||v|| for scalars α (absolute homogeneity), and (3) ||v+w|| ≤ ||v|| + ||w|| (triangle inequality).

A real-valued function on a vector space that assigns a nonnegative size to each element and satisfies positive definiteness (zero only at the zero vector), absolute homogeneity (scaling by scalars), and the triangle inequality (subadditivity).

A norm is a function ‖·‖ from a vector space to [0,∞) assigning each vector a nonnegative length and satisfying positive definiteness (‖v‖=0 iff v=0), absolute homogeneity (‖αv‖=|α|‖v‖), and the triangle inequality (‖u+v‖≤‖u‖+‖v‖).