Norm (mathematics)

In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—except for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector). A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below. In linear algebra, functional analysis, and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to each vector in a vector space—except for the zero vector, which is assigned a length of zero. A seminorm, on the other hand, is allowed to assign zero length to some non-zero vectors (in addition to the zero vector). A norm must also satisfy certain properties pertaining to scalability and additivity which are given in the formal definition below. A simple example is two dimensional Euclidean space R2 equipped with the 'Euclidean norm' (see below). Elements in this vector space (e.g., (3, 7)) are usually drawn as arrows in a 2-dimensional cartesian coordinate system starting at the origin (0, 0). The Euclidean norm assigns to each vector the length of its arrow. Because of this, the Euclidean norm is often known as the magnitude. A vector space on which a norm is defined is called a normed vector space. Similarly, a vector space with a seminorm is called a seminormed vector space. It is often possible to supply a norm for a given vector space in more than one way. Given a vector space V over a subfield F of the complex numbers, a norm on V is a nonnegative-valued scalar function p: V → For all a ∈ F and all u, v ∈ V, A seminorm on V is a function p : V → R with the properties 1 and 2 above. Every vector space V with seminorm p induces a normed space V/W, called the quotient space, where W is the subspace of V consisting of all vectors v in V with p(v) = 0. The induced norm on V/W is defined by: Two norms (or seminorms) p and q on a vector space V are equivalent if there exist two real constants c and C, with c > 0, such that A topological vector space is called normable (seminormable) if the topology of the space can be induced by a norm (seminorm).