Seminorm
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—save 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. 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 function p: V → R with the following properties:
For all a ∈ F and all u, v ∈ V,
There is some redundancy in this definition. By the absolute homogeneity axiom, we have p(0) = 0 and p(−v) = p(v), so that by the triangle inequality we get p(-v) + p(v) ≥ p(-v + v) = p(0) = 0, that is, p(v) ≥ 0. Thus, axioms 1 and 2 together imply axiom 3. However, axiom 4 is independent of the first three axioms.
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