Quantum states are elements of a complex Hilbert space $\mathcal{H}$ equipped with an inner product $\langle \phi|\psi\rangle$. Completeness ensures that convergent sequences of states have limits inside the space, enabling spectral analysis and expansion in bases.

The spectral theorem generalizes diagonalization. For self-adjoint operators, the operator can be represented through a projection-valued measure, allowing both discrete and continuous spectra to be treated on equal footing.

Continuous symmetries are represented by unitary operators $U(g)$ acting on $\mathcal{H}$. By Stone's theorem, strongly continuous one-parameter unitary groups are generated by self-adjoint operators.