Contents

group theory

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

1. $\rho(g) = 0$ if and only if $g$ is the identity element in $G$,
2. $\rho(g^{-1}) = \rho(g)$,
3. $\rho(g h) \le \rho(g) + \rho(h)$.
###### 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 $(G_1,\rho_1)\to (G_2,\rho_2)$ of normed groups, def. , is a group homomorphism $f \colon G_1 \to G_2$ of the underlying groups such that there is $C \in \mathbb{R}_{\geq 0}$ such that for all $g\in G_1$ we have $\rho_2(f(g)) \leq C\cdot\rho_1(g)$.

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

###### Remark

If $G$ is a vector space (viewed as an abelian group) the conditions on $\rho$ 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 $-1$ instead of for all elements of the coefficient field.

###### Remark

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

$d_L(g,h) = \rho(g^{-1} h), \qquad d_R(g,h) = \rho(g h^{-1})$

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 $(G,\rho)$ where $G = (G_1,G_2)$ is a groupoid and $\rho \colon G_2 \to [0,\infty)$ is a function on the arrows of $G$ satisfying the conditions:

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

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

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

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