Subrepresentation of a representation

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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.

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Subrepresentation of a representation. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by A.I. Shtern (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article