Subrepresentation of a representation
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.
Subrepresentation of a representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Subrepresentation_of_a_representation&oldid=48900