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 "L" 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 formula_1 from Lebesgue integration theory are complete. Modern forms of the theorem. The most common form of the theorem states that a measurable function on formula_2 is square integrable if and only if the corresponding Fourier series converges in  the Lp space formula_3 This means that if the "N"th partial sum of the Fourier series corresponding to a square-integrable function "f" is given by formula_4 where formula_5 the "n"th Fourier coefficient, is given by formula_6 then formula_7 where formula_8 is the formula_9-norm. Conversely, if formula_10 is a two-sided sequence of complex numbers (that is, its indices range from negative infinity to positive infinity) such that formula_11 then there exists a function "f" such that "f" is square-integrable and the values formula_12 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 formula_13 then formula_14 for all but countably many formula_15 and formula_16 Furthermore, if "A" is an orthonormal basis for "H" and "x" an arbitrary vector, the series formula_17 converges (or ) to "x". This is equivalent to saying that for every formula_18 there exists a finite set formula_19 in "A" such that formula_20 for every  finite set "B" containing "B". Moreover, the following conditions on the set "A" are equivalent: Another result, which also sometimes bears the name of Riesz and Fischer, is the theorem that formula_9 (or more generally formula_24) is complete. Example. 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 formula_25 be an orthonormal system in "R" (e.g. Fourier basis, Hermite or Laguerre polynomials, etc. – see orthogonal polynomials ), not necessarily complete (in an inner product space, an orthonormal set is complete if no nonzero vector is orthogonal to every vector in the set). The theorem asserts that if the normed space "R" is complete (thus "R" is a Hilbert space), then any sequence formula_26 that has finite formula_27 norm defines a function "f" in the space "R". The function "f" is defined by formula_28 limit in "R"-norm. Combined with the Bessel's inequality, we know the converse as well: if "f" is a function in "R", then the Fourier coefficients formula_29 have finite formula_27 norm. History: the Note  of Riesz and the Note of Fischer (1907). In his Note, states the following result (translated here to modern language at one point: the notation formula_31 was not used in 1907). Today, this result of Riesz is a special case of basic facts about series of orthogonal vectors in Hilbert spaces. Riesz's Note appeared in March. In May, states explicitly in a theorem (almost with modern words) that a Cauchy sequence in formula_31 converges in formula_9-norm to some function formula_39 In this Note, Cauchy sequences are called ""sequences converging in the mean"" and formula_31 is denoted by formula_41  Also, convergence to a limit in formula_9–norm is called ""convergence in the mean towards a function"". Here is the statement, translated from French: Fischer goes on proving the preceding result of Riesz, as a consequence of the orthogonality of the system, and of the completeness of formula_3 Fischer's proof of completeness is somewhat indirect. It uses the fact that the indefinite integrals of the functions "g" in the given Cauchy sequence, namely formula_46 converge uniformly on formula_47 to some function "G", continuous with bounded variation. The existence of the limit formula_48 for the Cauchy sequence is obtained by  applying to "G" differentiation theorems from Lebesgue's theory. Riesz uses a similar reasoning in his Note, but makes no explicit mention to the completeness of formula_49 although his result may be interpreted this way. He says that integrating term by term a trigonometric series with given square summable coefficients, he gets a series converging uniformly to a continuous function "F"  with bounded variation. The derivative "f"  of "F", defined almost everywhere, is square summable and has for "Fourier coefficients" the given coefficients. Completeness of "L",  0 < "p" ≤ ∞. For some authors, notably Royden, the Riesz-Fischer Theorem  is the result that formula_1 is complete: that every Cauchy sequence of functions in formula_1 converges to a function in formula_52 under the metric induced by the "p"-norm. The proof below is based on the convergence theorems for the Lebesgue integral; the result can also be obtained for formula_53 by showing that every Cauchy sequence has a rapidly converging Cauchy sub-sequence, that every Cauchy sequence with a convergent sub-sequence converges, and that every rapidly Cauchy sequence in formula_1 converges in formula_55 When formula_56 the Minkowski inequality implies that the Lp space formula_1 is a normed space. In order  to prove that formula_1 is complete, i.e. that formula_1 is a Banach space, it is enough (see e.g. Banach space#Definition) to prove that every series formula_60 of functions in formula_61 such that formula_62 converges in the formula_1-norm to some function formula_64 For formula_65 the Minkowski inequality and the monotone convergence theorem imply that formula_66 is defined formula_67–almost everywhere and formula_64 The dominated convergence theorem is then used to prove that the partial sums of the series converge to "f" in the formula_1-norm, formula_70 The case formula_71 requires some modifications, because the "p"-norm is no longer subadditive. One  starts with the stronger assumption that formula_72 and uses repeatedly that formula_73 The case formula_74 reduces to a simple question about uniform convergence outside a formula_67-negligible set.