PlanetMath: vector norm

(more info)

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vector norm

(Definition)

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

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

Such a function is denoted as $\vert\vert\,x\,\vert\vert$ . Particular norms are distinguished by subscripts, such as $\vert\vert\,x\,\vert\vert _V$ , when referring to a norm in the space . A unit vector with respect to the norm $\vert\vert\,\cdot\,\vert\vert$ is a vector satisfying $\vert\vert\,x\,\vert\vert = 1$ .

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 $\vert\vert x\vert\vert=(x_1^2 + x_2^2 + \cdots + x_n^2)^{1/2}$ 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

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