# Invariant subspace of a representation

From Encyclopedia of Mathematics

* of a group (algebra, ring, semi-group) in a vector space (or topological vector space) *

A vector (respectively, a closed vector) subspace such that for any and any one has . If is a projection operator from onto , then is an invariant subspace of if and only if for all . The subspace in is invariant for any representation in ; it is called the trivial invariant subspace; the remaining invariant subspaces (if there are any) are called non-trivial. See also Contraction of a representation; Completely-reducible set; Irreducible representation.

**How to Cite This Entry:**

Invariant subspace of a representation.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Invariant_subspace_of_a_representation&oldid=12674

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