HILBERT SPACES
Hilbert spaces [ edit ] Main article: Hilbert space The succeeding snapshots show summation of 1 to 5 terms in approximating a periodic function (blue) by finite sum of sine functions (red). Complete inner product spaces are known as Hilbert spaces , in honor of David Hilbert . [51] The Hilbert space � 2 ( Ω ) , with inner product given by ⟨ � , � ⟩ = ∫ Ω � ( � ) � ( � ) ¯ � � , where � ( � ) ¯ denotes the complex conjugate of � ( � ) , [52] [nb 11] is a key case. By definition, in a Hilbert space any Cauchy sequence converges to a limit. Conversely, finding a sequence of functions � � with desirable properties that approximates a given limit function, is equally crucial. Early analysis, in the guise of the Taylor approximation , established an approximation of differentiable functions � by polynomials. [53] By the Stone–Weierstrass theorem , ever...