# 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.