Definition:Norm/Vector Space

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Add an alternative (simplified) definition for vector spaces over usual number fields based on the Notes section. Prominently link to something about standard norms on real (and complex?) finite-dimensional vector spaces.
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This page is about Norm on Vector Space. For other uses, see Norm.

Definition

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 $

Division Ring

When the vector space $V$ is the $R$-vector space $R$, the definition reduces to the division ring norm:

Let $\struct {R, +, \circ}$ be a division ring whose zero is denoted $0_R$.

A (multiplicative) norm on $R$ is a mapping from $R$ to the non-negative reals:

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

satisfying the (ring) multiplicative norm axioms:

$(\text N 1)$	$:$	Positive Definiteness:	$\ds \forall x \in R:$	$\ds \norm x = 0 $	$\ds \iff $	$\ds x = 0_R $
$(\text N 2)$	$:$	Multiplicativity:	$\ds \forall x, y \in R:$	$\ds \norm {x \circ y} $	$\ds = $	$\ds \norm x \times \norm y $
$(\text N 3)$	$:$	Triangle Inequality:	$\ds \forall x, y \in R:$	$\ds \norm {x + y} $	$\ds \le $	$\ds \norm x + \norm y $

Notes

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In the literature, it is more common to define the norm only if $R$ is $\R$ or $\C$ (and consequently $\norm {\,\cdot\,}_R$ is the absolute value or modulus function respectively).

However, the definition given here incorporates this approach.

Also known as

The term length is occasionally seen as an alternative for norm.

Also see

Sources

2013: Francis Clarke: Functional Analysis, Calculus of Variations and Optimal Control ... (next): $1.1$: Basic Definitions
2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed spaces

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

\((\text N 1)\)	$:$	Positive Definiteness:	\(\ds \forall x \in R:\)	\(\ds \norm x = 0 \)	\(\ds \iff \)	\(\ds x = 0_R \)
\((\text N 2)\)	$:$	Multiplicativity:	\(\ds \forall x, y \in R:\)	\(\ds \norm {x \circ y} \)	\(\ds = \)	\(\ds \norm x \times \norm y \)
\((\text N 3)\)	$:$	Triangle Inequality:	\(\ds \forall x, y \in R:\)	\(\ds \norm {x + y} \)	\(\ds \le \)	\(\ds \norm x + \norm y \)