vector norm

A vector norm on the real vector space $V$ is a function $f:V\to\mathbb{R}$ that satisfies the following properties:

	$\displaystyle f(x)=0\iff x=0$
	$\displaystyle f(x)\geq 0$		$\displaystyle x\in V$
	$\displaystyle f(x+y)\leq f(x)+f(y)$		$\displaystyle x,y\in V$
	$\displaystyle f(\alpha x)=\|\alpha\|f(x)$		$\displaystyle\alpha\in\mathbb{R},x\in V$

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 vector with respect to the norm $||\,\cdot\,||$ 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 $||x||=(x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2})^{1/2}$ defined on $V=\mathbb{R}^{n}$ . Note, however, that there exists vector spaces 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

	$\displaystyle d(x+a,y+a)=$	$\displaystyle d(x,y)$	$\displaystyle\forall x,y,a\in V$
	$\displaystyle d(\alpha x,\alpha y)=$	$\displaystyle\|\alpha\|d(x,y)$	$\displaystyle\forall x,y\in V,\alpha\in\mathbb{R}$

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	SemiNorm
Related topic	BanachSpace
Related topic	HilbertSpace
Related topic	UnitVector
Defines	normed vector space
Defines	Euclidean norm