Norm (mathematics)

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

.