Complete inner product spaces are known as Hilbert spaces, in honor of David Hilbert. The Hilbert space L 2 ( Ω Ω ) , with inner product given by ⟨ ⟨ f , g ⟩ ⟩ = ∫ ∫ Ω Ω f ( x ) g ( x ) ¯ ¯ d x , where g ( x ) ¯ ¯ denotes the complex conjugate of g ( x ) , is a key case.