Proof. Let the wavefunctions ψ 1 , . . . , ψ K form an orthonormal basis of the eigenspace of Ĥ with eigenvalue ε, and the wavefunctions ψ ′ 1 , . . . , ψ ′ K be a distinct orthonormal basis of the same eigenspace. Since both sets of functions serve as a basis for the same functional space, there exists an irreducible unitary representation (norm-preserving map) R : G ε -→ U (K) that maps any element in the group G ε composed of all wavefunctions that are eigenfunctions of Ĥ, with eigenvalue ε, to a K × K unitary matrix that operates on a K-dimensional Hilbert space V (K), due to Ĥ being self-adjoint, such that: