Unitarization of a compact-group representation

For a compact group with normalized Haar measure, averaging any positive-definite Hermitian form,produces an invariant positive-definite Hermitian form.

A finite-dimensional complex representation of a compact group is a direct sum of irreducible representations. Average a positive Hermitian inner product over normalized Haar measure. The orthogonal complement of an invariant subspace is then invariant, so induction on dimension gives the decomposition. The same argument works for finite-dimensional real representations with a positive real inner product.

For a continuous real representation $\rho :G\to GL(V)$ of a compact group, average any positive-definite real inner product against normalized Haar measure:The result is positive definite and $G$-invariant. An orthonormal basis for it conjugates the representation into an orthogonal group.