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 functionf is given by where Fn, the nth Fourier coefficient, is given by then where is the L2-norm. Conversely, if is a two-sided sequence of complex numbers 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 spaceH, and x ∈ H, then for all but countably many y ∈ A, and Furthermore, if A is an orthonormal basis for H and x an arbitrary vector, the series converges commutatively to x. This is equivalent to saying that for every ε > 0, there exists a finite setB0 in A such that for every finite set B containing B0. 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 L2 is complete.
Example
The Riesz–Fischer theorem also applies in a more general setting. Let R be an inner product spaceconsisting of functions, and let be an orthonormal system in R, not necessarily complete. The theorem asserts that if the normed spaceR is complete, then any sequence that has finite ℓ2norm defines a function f in the space R. The function f is defined by , 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 have finite ℓ2 norm.
History: the Note of Riesz and the Note of Fischer (1907)
In his Note, states the following result. 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 that a Cauchy sequence in L2 converges in L2-norm to some function f in L2. In this Note, Cauchy sequences are called "sequences converging in the mean" and L2 is denoted by Ω. Also, convergence to a limit in L2-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 L2. Fischer's proof of completeness is somewhat indirect. It uses the fact that the indefinite integrals of the functions gn in the given Cauchy sequence, namely converge uniformly on to some function G, continuous with bounded variation. The existence of the limit g ∈ L2 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 L2, 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 functionF 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''''p'', 0 < ''p'' ≤ ∞
For some authors, notably Royden, the Riesz-Fischer Theorem is the result that Lp is complete: that every Cauchy sequence of functions in Lp converges to a function in Lp, 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 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 Lp converges in Lp. When 1 ≤ p ≤ ∞, the Minkowski inequality implies that the space Lp is a normed space. In order to prove that Lp is complete, i.e. that Lp is a Banach space, it is enough to prove that every series ∑ un of functions in Lp such that converges in the Lp-norm to some function f ∈ Lp. For p < ∞, the Minkowski inequality and the monotone convergence theorem imply that is defined μ-almost everywhere and f ∈ Lp. The dominated convergence theorem is then used to prove that the partial sums of the series converge to f in the Lp-norm, The case 0 < p < 1 requires some modifications, because the p-norm is no longer subadditive. One starts with the stronger assumption that and uses repeatedly that The case p = ∞ reduces to a simple question about uniform convergence outside a μ-negligible set.
