Quantum states live in a complex Hilbert space $\mathcal{H}$, where vectors are defined up to an overall phase. Physical predictions depend on inner products, norms, and linear maps, not on a chosen coordinate representation.

For a self-adjoint operator $A$, the spectral theorem generalizes diagonalization. In finite dimensions, $A=\sum_n a_n |a_n\rangle\langle a_n|$; in infinite dimensions, the spectral measure $E_A(\lambda)$ replaces a simple eigenbasis.

If $|\psi\rangle\in\mathcal{H}$, then the dual vector $\langle\psi|$ acts linearly on kets, giving amplitudes $\langle\phi|\psi\rangle$. The norm is $\|\psi\|^2=\langle\psi|\psi\rangle$ and probabilities follow the Born rule.

Momentum and position eigenstates are not normalizable in the usual Hilbert-space sense. They are handled rigorously using a rigged Hilbert space or Gelfand triple, where generalized eigenvectors act as distributions.