Norm (mathematics)

In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector. This norm can be defined as the square root of the inner product of a vector with itself.

A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin.^[1] A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space.

The term pseudonorm has been used for several related meanings. It may be a synonym of "seminorm".^[1] A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "${\displaystyle \,\leq \,}$" in the homogeneity axiom.^{[2][dubious – discuss]} It can also refer to a norm that can take infinite values,^[3] or to certain functions parametrised by a directed set.^[4]