Classical groups
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Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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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.
A normed group is a pair where is a group and is a function, the norm, satisfying the following conditions:
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 of normed groups, def. , is a group homomorphism of the underlying groups such that there is such that for all we have .
A short homomorphism of normed groups is a group homomorphism of the underlying groups such that for all we have .
If is a vector space (viewed as an abelian group) the conditions on 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 instead of for all elements of the coefficient field.
A norm on a group in def. defines two metrics:
The former is left invariant, the latter right invariant.
A normed group is not necessarily a topological group, see (Bingham-Ostaszweszki).
The definition can be extended to groupoids.
A normed groupoid is a pair where is a groupoid and is a function on the arrows of satisfying the conditions:
From a normed groupoid we do not just get a single metric space. Rather we get one metric space for each object. For the underlying set of the corresponding metric space is the set of all arrows with source . The metric is then . An arrow from to induces an isometry by right translation.
This reverses: from a metric space, say , we get a normed groupoid by taking the trivial groupoid on . An arrow in this groupoid is simply a pair of elements, whence we define the norm on by .
Last revised on July 13, 2014 at 13:45:36. See the history of this page for a list of all contributions to it.