Invariant subspace of a representation

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.