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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 "" in the homogeneity axiom.[2] It can also refer to a norm that can take infinite values,[3] or to certain functions parametrised by a directed set.[4]
Definition
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 :[5]
- Subadditivity/Triangle inequality: for all
- Absolute homogeneity: for all and all scalars
- Positive definiteness/positiveness[6]/Point-separating: 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.)[7] 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:[6] for all
Some authors include non-negativity as part of the definition of "norm", although this is not necessary. Although this article defined "positive" to be a synonym of "positive definite", some authors instead define "positive" to be a synonym of "non-negative";[8] these definitions are not equivalent.
Equivalent norms
Suppose that and are two norms (or seminorms) on a vector space Then and are called equivalent, if there exist two positive real constants and with such that for every vector
Notation
If a norm is given on a vector space then the norm of a vector is usually denoted by enclosing it within double vertical lines:
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