# Definition:Norm/Vector Space

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

## Contents

## 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: | \(\displaystyle \forall x, y \in V:\) | \(\displaystyle \norm {x + y} \) | \(\displaystyle \le \) | \(\displaystyle \norm x + \norm y \) |

### 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**:

\((\text N 1)\) | $:$ | 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**.

## Also see

- Definition:Norm on Division Ring
- Definition:Norm on Algebra
- Definition:Norm/Bounded Linear Transformation
- Definition:Norm/Bounded Linear Functional

## 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*: Chapter $1$: Normed and Banach spaces