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 or length 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". It can also refer to a norm that can take infinite values or to certain functions parametrised by a directed set.
Definition Given a vector space X {\displaystyle X} over a subfield F {\displaystyle F} of the complex numbers C , {\displaystyle \mathbb {C} ,} a norm on X {\displaystyle X} is a real-valued function p : X → R {\displaystyle p:X\to \mathbb {R} } with the following properties, where | s | {\displaystyle |s|} denotes the usual absolute value of a scalar s {\displaystyle s} :
Subadditivity / Triangle inequality: p ( x + y ) ≤ p ( x ) + p ( y ) {\displaystyle p(x+y)\leq p(x)+p(y)} for all x , y ∈ X . {\displaystyle x,y\in X.}
Absolute homogeneity: p ( s x ) = | s | p ( x ) {\displaystyle p(sx)=|s|p(x)} for all x ∈ X {\displaystyle x\in X} and all scalars s . {\displaystyle s.}
Positive definiteness / Positiveness / Point-separating:for all x ∈ X , {\displaystyle x\in X,} if p ( x ) = 0 , {\displaystyle p(x)=0,} then x = 0. {\displaystyle x=0.}
Because property (2.) implies p ( 0 ) = 0 , {\displaystyle p(0)=0,} some authors replace property (3.) with the equivalent condition: for every x ∈ X , {\displaystyle x\in X,} p ( x ) = 0 {\displaystyle p(x)=0} if and only if x = 0. {\displaystyle x=0.}
A seminorm on X {\displaystyle X} is a function p : X → R {\displaystyle p:X\to \mathbb {R} } 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 p {\displaystyle p} is a norm (or more generally, a seminorm), then p ( 0 ) = 0 {\displaystyle p(0)=0} and that p {\displaystyle p} also has the following property:
Non-negativity: p ( x ) ≥ 0 {\displaystyle p(x)\geq 0} for all x ∈ X . {\displaystyle x\in X.}
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"; these definitions are not equivalent.
Notation If a norm p : X → R {\displaystyle p:X\to \mathbb {R} } is given on a vector space X , {\displaystyle X,} then the norm of a vector z ∈ X {\displaystyle z\in X} is usually denoted by enclosing it within double vertical lines: ‖ z ‖ = p ( z ) {\displaystyle \|z\|=p(z)} , as proposed by Stefan Banach in his doctoral thesis from 1920. Such notation is also sometimes used if p {\displaystyle p} is only a seminorm. For the length of a vector in Euclidean space (which is an example of a norm, as explained below), the notation | x | {\displaystyle |x|} with single vertical lines is also widespread.
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