A.3.1 Definition

A norm is a real-valued function $\Vert\cdot\Vert$ on a linear space $X \subseteq \mathbb{R}^n$ such that

$\displaystyle \Vert{\mathbf{{x}}}+{\mathbf{{y}}}\Vert \quad$		$\displaystyle \leq \quad \Vert{\mathbf{{x}}}\Vert+\Vert{\mathbf{{y}}}\Vert$	(A.5)
$\displaystyle \Vert\alpha{\mathbf{{x}}}\Vert \quad$		$\displaystyle = \quad \vert\alpha\vert \Vert{\mathbf{{x}}}\Vert$	(A.6)

$\displaystyle \Vert{\mathbf{{x}}}\Vert {=}0\quad \Longleftrightarrow \quad{\mathbf{{x}}}{=}\bf {0}\quad\quad$

(A.7)

where ${\mathbf{{x}}},{\mathbf{{y}}}\in X$ and $\alpha \in \mathbb{R}$ .