# Definition:Norm/Vector Space

## 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: $\displaystyle \forall x \in V:$ $\displaystyle \norm x = 0$ $\displaystyle \iff$ $\displaystyle x = \mathbf 0_V$ $(\text N 2)$ $:$ 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: $\displaystyle \forall x, y \in V:$ $\displaystyle \norm {x + y}$ $\displaystyle \le$ $\displaystyle \norm x + \norm y$

### 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: $\displaystyle \forall x \in R:$ $\displaystyle \norm x = 0$ $\displaystyle \iff$ $\displaystyle x = 0_R$ $(\text N 2)$ $:$ Multiplicativity: $\displaystyle \forall x, y \in R:$ $\displaystyle \norm {x \circ y}$ $\displaystyle =$ $\displaystyle \norm x \times \norm y$ $(\text N 3)$ $:$ 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.