Invariant subspace of a representation

$ \pi $ of a group (algebra, ring, semi-group) $ X $ in a vector space (or topological vector space) $ E $

A vector (respectively, a closed vector) subspace $ F \subset E $ such that for any $ \xi \in F $ and any $ x \in X $ one has $ \pi ( x ) \xi \in F $. If $ P $ is a projection operator from $ E $ onto $ F $, then $ F $ is an invariant subspace of $ \pi $ if and only if $ P \pi ( x ) P = \pi ( x ) P $ for all $ x \in X $. The subspace $ \{ 0 \} $ in $ E $ is invariant for any representation in $ E $; 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.