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 that has finite norm 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 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 was not used in 1907). Let be an orthonormal system in and a sequence of reals. The convergence of the series is a necessary and sufficient condition for the existence of a function f such that 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 converges in -norm to some function In this Note, Cauchy sequences are called "sequences converging in the mean" and is denoted by Also, convergence to a limit in –norm is called "convergence in the mean towards a function". Here is the statement, translated from French: Theorem. If a sequence of functions belonging to converges in the mean, there exists in a function f towards which the sequence converges in the mean. Fischer goes on proving the preceding result of Riesz, as a consequence of the orthogonality of the system, and of the completeness of 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 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 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