Riesz–Fischer theorem

In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer. In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer. For many authors, the Riesz–Fischer theorem refers to the fact that the Lp spaces from Lebesgue integration theory are complete. The most common form of the theorem states that a measurable function on is square integrable if and only if the corresponding Fourier series converges in the space L2. This means that if the Nth partial sum of the Fourier series corresponding to a square-integrable function f is given by where Fn, the nth Fourier coefficient, is given by