nLab
norm

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Definition

On an abelian group

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

||:G {\vert-\vert} \;\colon\; G \longrightarrow \mathbb{R}

to the real numbers, such that

  1. (positivity) (g0)(|g|>0)(g \neq 0) \Rightarrow (\vert g\vert \gt 0)

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

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

Here |k|{\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 kk 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 kk-vector space VV is a function

||:V {\vert-\vert} \colon V \to \mathbb{R}

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

  1. |λv|=|λ||v|{\vert \lambda v \vert} = {\vert \lambda\vert} {\vert v \vert} (where |λ|\vert \lambda \vert denotes the valuation)

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

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

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

If the triangle identity is strengthened to

  • |v+w|max(|v|,|w|){\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 || 1{\vert - \vert}_1 and || 2{\vert - \vert}_2 are called equivalent if there are 0<C,C0 \lt C, C' \in \mathbb{R} such that for all vv we have

C|v| 1|v| 2C|v| 1. 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 n\mathbb{R}^n the Euclidean norm is given by

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

Minkowski Functionals

Let VV be a vector space and BVB \subseteq V an absorbing absolutely convex subset. The Minkowski functional of BB is the function μ B:V\mu_B \colon V \to \mathbb{R} defined by:

μ B(v)=inf{t>0:vtB} \mu_B(v) = \inf\{t \gt 0 : v \in t B\}

This is a semi-norm on VV.

Properties

algebraic structuregroupringfieldvector spacealgebra
(submultiplicative) normnormed groupnormed ringnormed fieldnormed vector spacenormed algebra
multiplicative norm (absolute value/valuation)valued field
completenesscomplete normed groupBanach ringcomplete fieldBanach vector spaceBanach algebra

References

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