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'' $  \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|Contraction of a representation]]; [[Completely-reducible set|Completely-reducible set]]; [[Irreducible representation|Irreducible representation]].