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. 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 Lp space This means that if the Nth partial sum of the Fourier series corresponding to a square-integrable function f is given by where the nth Fourier coefficient, is given by then where is the -norm. Conversely, if is a two-sided sequence of complex numbers (that is, its indices range from negative infinity to positive infinity) such that then there exists a function f such that f is square-integrable and the values are the Fourier coefficients of f. This form of the Riesz–Fischer theorem is a stronger form of Bessel's inequality, and can be used to prove Parseval's identity for Fourier series. Other results are often called the Riesz–Fischer theorem . Among them is the theorem that, if A is an orthonormal set in a Hilbert space H, and then for all but countably many and Furthermore, if A is an orthonormal basis for H and x an arbitrary vector, the series converges (or ) to x. This is equivalent to saying that for every there exists a finite set in A such that for every finite set B containing B0. Moreover, the following conditions on the set A are equivalent: the set A is an orthonormal basis of H for every vector Another result, which also sometimes bears the name of Riesz and Fischer, is the theorem that (or more generally ) is complete. The Riesz–Fischer theorem also applies in a more general setting. Let R be an inner product space consisting of functions (for example, measurable functions on the line, analytic functions in the unit disc; in old literature, sometimes called Euclidean Space), and let be an orthonormal system in R (e.