# Contents

## Definition

For $k$ a field equipped with a valuation (most usually, a local field such as $ℝ$, $ℂ$, or a p-adic completion of a number field), a norm on a $k$-vector space $V$ is a function

$\mid -\mid :V\to ℝ${\vert-\vert} : V \to \mathbb{R}

such that for all $\lambda \in k$, $v,w\in V$ we have

1. $\mid \lambda v\mid =\mid \lambda \mid \mid v\mid$ (where $\mid \lambda \mid$ denotes the valuation)

2. $\mid v+w\mid \le \mid v\mid +\mid w\mid$ (“triangle inequality?”)

3. if $\mid v\mid =0$ then $v=0$.

If the third property is not required, one speaks of a seminorm.

If the triangle identity is strengthened to

• $\mid v+w\mid \le \mathrm{max}\left(\mid v\mid ,\mid w\mid \right)$

one speaks of a non-archimedean seminorm.

A vector space equipped with a norm is a normed vector space.

Each seminorm determines a topology, which is Hausdorff precisely if it is a norm.

A topological vector space is called (semi-)normed if its topology can be induced by a (semi-)norm.

Two seminorms ${\mid -\mid }_{1}$ and ${\mid -\mid }_{2}$ are called equivalent if there are $0 such that for all $v$ we have

$C{\mid v\mid }_{1}\le {\mid v\mid }_{2}\le C\prime {\mid v\mid }_{1}\phantom{\rule{thinmathspace}{0ex}}.$C {\vert v \vert}_1 \leq {\vert v \vert}_2 \leq C' {\vert v \vert}_1 \,.

Equivalent seminorms determine the same topology.

The collection of (bounded) multiplicative seminorms on a (Banach) ring is called its analytic spectrum (see there for details).

## Examples

• The standard absolute value is a norm on the real numbers.

• More generally, on any Cartesian space ${ℝ}^{n}$ the Euclidean norm is given by

$\left({x}^{1},\cdots ,{x}^{n}\right)↦\sqrt{\left(}\sum _{i=1}^{n}\left({x}^{i}{\right)}^{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$(x^1, \cdots, x^n) \mapsto \sqrt(\sum_{i=1}^n (x^i)^2) \,.

## Minkowski Functionals

Let $V$ be a vector space and $B\subseteq V$ an absorbing absolutely convex subset. The Minkowski functional of $B$ is the function ${\mu }_{B}:V\to ℝ$ defined by:

${\mu }_{B}\left(v\right)=\mathrm{inf}\left\{t>0:v\in tB\right\}$\mu_B(v) = \inf\{t \gt 0 : v \in t B\}

This is a semi-norm on $V$.

Revised on May 23, 2013 18:41:23 by Andrew Stacey (192.76.7.217)