In the basis where X is diagonal, an arbitrary state can be written as a superposition of states with eigenvalues x, | ψ ψ ⟩ ⟩ = ∫ ∫ x ψ ψ ( x ) | x ⟩ ⟩ , so that ψ (x) = ⟨ x | ψ ⟩, and the operator X multiplies each eigenvector by x, X | ψ ψ ⟩ ⟩ = ∫ ∫ x x ψ ψ ( x ) | x ⟩ ⟩ .