nLab norm

Context

Analytic geometry

analytic geometry (complex, rigid, global)

Basic concepts

analytic function

analytification

GAGA

Contents

Definition

On an abelian group

For $(A,+)$ an abelian group, then a norm on the group is a function

${\vert-\vert} \;\colon\; G \longrightarrow \mathbb{R}$

to the real numbers, such that

1. (positivity) $(g \neq 0) \Rightarrow (\vert g\vert \gt 0)$

2. (triangle inequality) ${\vert g + h\vert}\leq {\vert g\vert} + {\vert h\vert}$

3. (linearity) ${\vert k g\vert} = {\vert k\vert} {\vert g\vert}$ for all $k \in \mathbb{Z}$.

Here ${\vert k\vert} \in \mathbb{N}$ denotes the absolute value.

A group with a norm is a normed group, see there for more.

On a vector space

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

${\vert-\vert} \colon V \to \mathbb{R}$

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

1. ${\vert \lambda v \vert} = {\vert \lambda\vert} {\vert v \vert}$ (where $\vert \lambda \vert$ denotes the valuation)

2. ${\vert v + w\vert } \leq {\vert v \vert } + {\vert w \vert}$ (“triangle inequality”)

3. if ${\vert v\vert} = 0$ then $v = 0$.

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

If the triangle identity is strengthened to

• ${\vert v + w\vert } \leq max ({\vert v\vert}, {\vert w\vert})$

one speaks of a non-archimedean seminorm, otherwise of an archimedean one.

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 ${\vert - \vert}_1$ and ${\vert - \vert}_2$ are called equivalent if there are $0 \lt C, C' \in \mathbb{R}$ such that for all $v$ we have

$C {\vert v \vert}_1 \leq {\vert v \vert}_2 \leq C' {\vert v \vert}_1 \,.$

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

General

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

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

$(x^1, \cdots, x^n) \mapsto \sqrt(\sum_{i=1}^n (x^i)^2) \,.$

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 $\mu_B \colon V \to \mathbb{R}$ defined by:

$\mu_B(v) = \inf\{t \gt 0 : v \in t B\}$

This is a semi-norm on $V$.

References

Revised on July 18, 2014 04:54:28 by Urs Schreiber (82.136.246.44)