Definition:Norm/Vector Space

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This page is about the norm on a vector space. For other uses, see Definition: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:

\((N1)\)   $:$   Positive definiteness:      \(\displaystyle \forall x \in V:\)    \(\displaystyle \norm x = 0 \)   \(\displaystyle \iff \)   \(\displaystyle x = \mathbf 0_V \)             
\((N2)\)   $:$   Positive homogeneity:      \(\displaystyle \forall x \in V, \lambda \in R:\)    \(\displaystyle \norm {\lambda x} \)   \(\displaystyle = \)   \(\displaystyle \norm {\lambda}_R \times \norm x \)             
\((N3)\)   $:$   Triangle inequality:      \(\displaystyle \forall x, y \in V:\)    \(\displaystyle \norm {x + y} \)   \(\displaystyle \le \)   \(\displaystyle \norm x + \norm y \)             


Normed Vector Space

Let $\norm {\,\cdot\,}$ be a norm on $V$.


Then $\struct {V, \norm {\,\cdot\,} }$ is a normed vector space.


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:

\((N1)\)   $:$   Positive Definiteness:      \(\displaystyle \forall x \in R:\)    \(\displaystyle \norm x = 0 \)   \(\displaystyle \iff \)   \(\displaystyle x = 0_R \)             
\((N2)\)   $:$   Multiplicativity:      \(\displaystyle \forall x, y \in R:\)    \(\displaystyle \norm {x \circ y} \)   \(\displaystyle = \)   \(\displaystyle \norm x \times \norm y \)             
\((N3)\)   $:$   Triangle Inequality:      \(\displaystyle \forall x, y \in R:\)    \(\displaystyle \norm {x + y} \)   \(\displaystyle \le \)   \(\displaystyle \norm x + \norm y \)             


Notes

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