In quantum theory, it is also possible for an operator to have a non-discrete spectrum, such as the position operator X in quantum mechanics. This operator has a completely continuous spectrum, with eigenvalues and eigenvectors depending on a continuous parameter, x . Specifically, the operator X acts on a spatial vector | x ⟩ ⟩ as X | x ⟩ ⟩ = x | x ⟩ ⟩ . In this case, the vector ψ ψ can be written as a complex-valued function ψ ψ ( x ) on the spectrum of X (usually the real line). This is formally achieved by projecting the state vector | ψ ψ ⟩ ⟩ onto the eigenvalues of the operator, as in the discrete case ψ ψ ( x ) ≡ ≡ ⟨ ⟨ x | ψ ψ ⟩ ⟩ . It happens that the eigenvectors of the position operator form a complete basis for the vector space of states, and therefore obey a completeness relation in quantum mechanics : ∫ ∫ | x ⟩ ⟩ ⟨ ⟨ x | d x ≡ ≡ I