Category:Quasinorms

This category contains results about Quasinorms.
Definitions specific to this category can be found in Definitions/Quasinorms.

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

Let $V$ be a vector space over $R$.

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

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

$(\text Q 1)$	$:$		Positive Homogeneity:	$\ds \forall x \in V, \lambda \in R:$		$\ds \norm {\lambda x} $	$\ds = $	$\ds \norm {\lambda}_R \times \norm x $
$(\text Q 2)$	$:$		Weak Triangle Inequality:	$\ds \exists M \ge 1, \forall x, y \in V:$		$\ds \norm {x + y} $	$\ds \le $	$\ds M \paren {\norm x + \norm y} $

Pages in category "Quasinorms"

This category contains only the following page.

\((\text Q 1)\)	$:$		Positive Homogeneity:	\(\ds \forall x \in V, \lambda \in R:\)		\(\ds \norm {\lambda x} \)	\(\ds = \)	\(\ds \norm {\lambda}_R \times \norm x \)
\((\text Q 2)\)	$:$		Weak Triangle Inequality:	\(\ds \exists M \ge 1, \forall x, y \in V:\)		\(\ds \norm {x + y} \)	\(\ds \le \)	\(\ds M \paren {\norm x + \norm y} \)