Symbols:Analysis/Norm

Norm

$\norm z$

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:	\(\ds \forall x \in V:\)		\(\ds \norm x = 0 \)	\(\ds \iff \)	\(\ds x = \mathbf 0_V \)
\((\text N 2)\)	$:$		Positive Homogeneity:	\(\ds \forall x \in V, \lambda \in R:\)		\(\ds \norm {\lambda x} \)	\(\ds = \)	\(\ds \norm {\lambda}_R \times \norm x \)
\((\text N 3)\)	$:$		Triangle Inequality:	\(\ds \forall x, y \in V:\)		\(\ds \norm {x + y} \)	\(\ds \le \)	\(\ds \norm x + \norm y \)

The $\LaTeX$ code for \(\norm z\) is \norm z .

Sources

• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Appendix: Table $7$: Common signs and symbols: norm
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $14$: Symbols