nLab
normed group

Contents

Context

Group Theory

Analysis

Contents

Idea

A normed group is to a group what a normed vector space is to a vector space. It consists of a group together with a length function (a norm) and, as for normed vector spaces, gives rise to a metric space.

A complete normed group is a complete normed group.

Definition

Normed groups

Definition

A normed group is a pair (G,ρ)(G,\rho) where GG is a group and ρ:G[0,)\rho \colon G \to [0,\infty) \subset \mathbb{R} is a function, the norm, satisfying the following conditions:

  1. ρ(g)=0\rho(g) = 0 if and only if gg is the identity element in GG,
  2. ρ(g 1)=ρ(g)\rho(g^{-1}) = \rho(g),
  3. ρ(gh)ρ(g)+ρ(h)\rho(g h) \le \rho(g) + \rho(h).
Definitions

There are a few different senses of homomorphism between normed groups; the first is usually taken as the default but the second fits in better with normed rings and the obvious notion of isomorphism as two structures' being ‘the same’.

A bounded homomorphism (G 1,ρ 1)(G 2,ρ 2)(G_1,\rho_1)\to (G_2,\rho_2) of normed groups, def. , is a group homomorphism f:G 1G 2f \colon G_1 \to G_2 of the underlying groups such that there is C 0C \in \mathbb{R}_{\geq 0} such that for all gG 1g\in G_1 we have ρ 2(f(g))Cρ 1(g)\rho_2(f(g)) \leq C\cdot\rho_1(g).

A short homomorphism (G 1,ρ 1)(G 2,ρ 2)(G_1,\rho_1)\to (G_2,\rho_2) of normed groups is a group homomorphism f:G 1G 2f \colon G_1 \to G_2 of the underlying groups such that for all gG 1g\in G_1 we have ρ 2(f(g))ρ 1(g)\rho_2(f(g)) \leq \cdot\rho_1(g).

Remark

If GG is a vector space (viewed as an abelian group) the conditions on ρ\rho in def. almost correspond to the axioms for a norm in the context of a normed vector space. The difference is that homogeneity is only assumed for 1-1 instead of for all elements of the coefficient field.

Remark

A norm on a group in def. defines two metrics:

d L(g,h)=ρ(g 1h),d R(g,h)=ρ(gh 1) d_L(g,h) = \rho(g^{-1} h), \qquad d_R(g,h) = \rho(g h^{-1})

The former is left invariant, the latter right invariant.

Remark

A normed group is not necessarily a topological group, see (Bingham-Ostaszweszki).

Normed groupoids

The definition can be extended to groupoids.

Definition

A normed groupoid is a pair (G,ρ)(G,\rho) where G=(G 1,G 2)G = (G_1,G_2) is a groupoid and ρ:G 2[0,)\rho \colon G_2 \to [0,\infty) is a function on the arrows of GG satisfying the conditions:

  1. ρ(g)=0\rho(g) = 0 if and only if gg is an identity arrow,
  2. ρ(g)=ρ(g 1)\rho(g) = \rho(g^{-1}),
  3. if ghg h exists then ρ(gh)ρ(g)+ρ(h)\rho(g h) \le \rho(g) + \rho(h).

From a normed groupoid we do not just get a single metric space. Rather we get one metric space for each object. For xG 1x \in G_1 the underlying set of the corresponding metric space is the set of all arrows with source xx. The metric is then d x(g,h)=ρ(gh 1)d_x(g,h) = \rho(g h^{-1}). An arrow from xx to yy induces an isometry by right translation.

This reverses: from a metric space, say XX, we get a normed groupoid by taking the trivial groupoid on XX. An arrow in this groupoid is simply a pair (x,y)(x,y) of elements, whence we define the norm on GX×XG \coloneqq X \times X by ρ(x,y)=d X(x,y)\rho(x,y) = d_X(x,y).

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

  • N. Bingham, A. Ostaszewski, Normed groups: dichotomy and duality (, pdf, pdf)

Last revised on July 13, 2014 at 13:45:36. See the history of this page for a list of all contributions to it.