Category:Definitions/Examples of Norms

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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:

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

satisfying the (vector space) norm axioms:

\((\text N 1)\)   $:$   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:      \(\displaystyle \forall x, y \in V:\)    \(\displaystyle \norm {x + y} \)   \(\displaystyle \le \)   \(\displaystyle \norm x + \norm y \)