Subrepresentation

In representation theory, a subrepresentation of a representation formula_1 of a group "G" is a representation formula_2 such that "W" is a vector subspace of "V" and formula_3.
A finite-dimensional representation always contains a nonzero subrepresentation that is irreducible, the fact seen by induction on dimension. This fact is generally false for infinite-dimensional representations.
If formula_1 is a representation of "G", then there is the trivial subrepresentation: