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. 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". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality "" in the homogeneity axiom. It can also refer to a norm that can take infinite values, or to certain functions parametrised by a directed set. Given a vector space over a subfield of the complex numbers a norm on is a real-valued function with the following properties, where denotes the usual absolute value of a scalar : Subadditivity/Triangle inequality: for all Absolute homogeneity: for all and all scalars Positive definiteness/positiveness/: for all if then Because property (2.) implies some authors replace property (3.) with the equivalent condition: for every if and only if A seminorm on is a function that has properties (1.) and (2.) so that in particular, every norm is also a seminorm (and thus also a sublinear functional). However, there exist seminorms that are not norms. Properties (1.) and (2.) imply that if is a norm (or more generally, a seminorm) then and that also has the following property: Non-negativity: for all Some authors include non-negativity as part of the definition of "norm", although this is not necessary.