vector norm

A vector normMathworldPlanetmath on the real vector space V is a function f:V that satisfies the following properties:

f(x)0 xV
f(x+y)f(x)+f(y) x,yV
f(αx)=|α|f(x) α,xV

Such a function is denoted as ||x||. Particular norms are distinguished by subscripts, such as ||x||V, when referring to a norm in the space V. A unit vectorMathworldPlanetmath with respect to the norm |||| is a vector x satisfying ||x||=1.

A vector norm on a complex vector space is defined similarly.

A common (and useful) example of a real norm is the Euclidean norm given by ||x||=(x12+x22++xn2)1/2 defined on V=n. Note, however, that there exists vector spacesMathworldPlanetmath which are metric, but upon which it is not possible to define a norm. If it possible, the space is called a normed vector space. Given a metric on the vector space, a necessary and sufficient condition for this space to be a normed space, is

d(x+a,y+a)= d(x,y) x,y,aV
d(αx,αy)= |α|d(x,y) x,yV,α

But given a norm, a metric can always be defined by the equation d(x,y)=||x-y||. Hence every normed space is a metric space.

Title vector norm
Canonical name VectorNorm
Date of creation 2013-03-22 11:43:00
Last modified on 2013-03-22 11:43:00
Owner mike (2826)
Last modified by mike (2826)
Numerical id 30
Author mike (2826)
Entry type Definition
Classification msc 46B20
Classification msc 18-01
Classification msc 20H15
Classification msc 20B30
Related topic Vector
Related topic Metric
Related topic Norm
Related topic VectorPnorm
Related topic NormedVectorSpace
Related topic MatrixNorm
Related topic MatrixPnorm
Related topic FrobeniusMatrixNorm
Related topic CauchySchwarzInequality
Related topic MetricSpace
Related topic VectorSpace
Related topic LpSpace
Related topic OperatorNorm
Related topic BoundedOperator
Related topic SemiNormMathworldPlanetmath
Related topic BanachSpace
Related topic HilbertSpace
Related topic UnitVector
Defines normed vector space
Defines Euclidean norm