Editing Unitary representation (section)

==Complete reducibility==

A unitary representation is [[Semisimple representation|completely reducible]], in the sense that for any closed [[invariant subspace]], the [[orthogonal complement]] is again a closed invariant subspace. This is at the level of an observation, but is a fundamental property. For example, it implies that finite-dimensional unitary representations are always a direct sum of irreducible representations, in the algebraic sense.

Since unitary representations are much easier to handle than the general case, it is natural to consider '''unitarizable representations''', those that become unitary on the introduction of a suitable complex Hilbert space structure. This works very well for [[representations of a finite group|finite group]]s, and more generally for [[compact group]]s, by an averaging argument applied to an arbitrary hermitian structure (more specifically, a new inner product defined by an averaging argument over the old one, w.r.t which the representation is unitary).<ref>{{harvnb|Hall|2015}} Section 4.4</ref> For example, a natural proof of [[Maschke's theorem]] is by this route.