Difference between revisions of "Subrepresentation of a representation"
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| + | A [[Linear representation|linear representation]] $ \rho $ | ||
| + | in an invariant subspace $ F \subset E $ | ||
| + | of a representation $ \pi $ | ||
| + | of a group (algebra, ring or semi-group) $ X $ | ||
| + | in a (topological) vector space $ E $ | ||
| + | defined by the formula $ \rho ( x) \xi = \pi ( x) \xi $ | ||
| + | for all $ \xi \in F $, | ||
| + | $ x \in X $. | ||
| + | If $ \pi $ | ||
| + | is a [[Continuous representation|continuous representation]] (of a topological group, algebra, ring, or semi-group), then any subrepresentation of it is also continuous. | ||
Latest revision as of 08:24, 6 June 2020
A linear representation $ \rho $
in an invariant subspace $ F \subset E $
of a representation $ \pi $
of a group (algebra, ring or semi-group) $ X $
in a (topological) vector space $ E $
defined by the formula $ \rho ( x) \xi = \pi ( x) \xi $
for all $ \xi \in F $,
$ x \in X $.
If $ \pi $
is a continuous representation (of a topological group, algebra, ring, or semi-group), then any subrepresentation of it is also continuous.
How to Cite This Entry:
Subrepresentation of a representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subrepresentation_of_a_representation&oldid=48900
Subrepresentation of a representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subrepresentation_of_a_representation&oldid=48900
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article