Norm (mathematics)

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This editable Main Article is under development and subject to a disclaimer.

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In mathematics, a norm is a function on a vector space that generalizes to vector spaces the notion of the distance from a point of a Euclidean space to the origin.

## Formal definition of norm[edit]

Let X be a vector space over some subfield F of the complex numbers. Then a norm on X is any function  ‖ ⋅ ‖ : X -> R {\displaystyle \|\cdot \|:X\rightarrow \mathbb {R} } having the following four properties:

1. ‖ x ‖ ≥ 0 {\displaystyle \|x\|\geq 0} for all  x ∈ X {\displaystyle x\in X} (positivity)
2. ‖ x ‖ = 0 {\displaystyle \|x\|=0} if and only if x=0
3. ‖ x + y ‖ ≤ ‖ x ‖ + ‖ y ‖ {\displaystyle \|x+y\|\leq \|x\|+\|y\|} for all  x , y ∈ X {\displaystyle x,y\in X} (triangular inequality)
4. ‖ c x ‖ = | c | ‖ x ‖ {\displaystyle \|cx\|=|c|\|x\|} for all  c ∈ F {\displaystyle c\in F}

A norm on X also defines a metric d {\displaystyle d} on X as  d ( x , y ) = ‖ x − y ‖ {\displaystyle d(x,y)=\|x-y\|} . Hence a normed space is also a metric space.