Category:Definitions/Examples of Norms

This category contains definitions of examples of norms.

Let $\struct {R, +, \circ}$ be a division ring with norm $\norm {\,\cdot\,}_R$.

Let $V$ be a vector space over $R$, with zero $0_V$.

A norm on $V$ is a map from $V$ to the nonnegative reals:

$\norm{\,\cdot\,}: V \to \R_{\ge 0}$

$(\text N 1)$	$:$	Positive definiteness:	$\displaystyle \forall x \in V:$	$\displaystyle \norm x = 0 $	$\displaystyle \iff $	$\displaystyle x = \mathbf 0_V $
$(\text N 2)$	$:$	Positive homogeneity:	$\displaystyle \forall x \in V, \lambda \in R:$	$\displaystyle \norm {\lambda x} $	$\displaystyle = $	$\displaystyle \norm {\lambda}_R \times \norm x $
$(\text N 3)$	$:$	Triangle inequality:	$\displaystyle \forall x, y \in V:$	$\displaystyle \norm {x + y} $	$\displaystyle \le $	$\displaystyle \norm x + \norm y $

Subcategories

This category has the following 2 subcategories, out of 2 total.

The following 6 pages are in this category, out of 6 total.

\((\text N 1)\)	$:$	Positive definiteness:	\(\displaystyle \forall x \in V:\)	\(\displaystyle \norm x = 0 \)	\(\displaystyle \iff \)	\(\displaystyle x = \mathbf 0_V \)
\((\text N 2)\)	$:$	Positive homogeneity:	\(\displaystyle \forall x \in V, \lambda \in R:\)	\(\displaystyle \norm {\lambda x} \)	\(\displaystyle = \)	\(\displaystyle \norm {\lambda}_R \times \norm x \)
\((\text N 3)\)	$:$	Triangle inequality:	\(\displaystyle \forall x, y \in V:\)	\(\displaystyle \norm {x + y} \)	\(\displaystyle \le \)	\(\displaystyle \norm x + \norm y \)