The function f ( x ) = x ⋅ ⋅ ψ ψ ( x ) can be interpreted as a vector in a function space. We can define an inner product for a pair of functions u (x) and v (x) in this vector space: ⟨ ⟨ u ∣ ∣ v ⟩ ⟩ = ∫ ∫ − − ∞ ∞ ∞ ∞ u ∗ ∗ ( x ) ⋅ ⋅ v ( x ) d x , where the asterisk denotes the complex conjugate.