Talk:Riesz–Fischer theorem

In the interest of un-stubbing this article, I ask others to include in it any applications of the Riesz-Fischer theorem. I will do the same. NatusRoma 21:16, 13 May 2005 (UTC)

convergence requirement
In Kolmogorov and Fomin, Introductory Real Analysis, ISBN 0486612260, page 153 says the following:  Given a Euclidean space R, let {φk} be an orthonormal (but not necessarily complete) system in R. It follows from Bessel's inequality that a necessary condition for the numbers c1,c2,…ck,… to be Fourier coefficients of an element f ∈ R is that the series
 * $$\sum_{k=1}^\infty c_k^2$$

converge. It turns out that this condition is also sufficient if R is complete, as shown by
 * Theorem 9 (Riesz-Fischer). Given an orthonormal system {&phi;k} in a complete Euclidean space R, let the numbers c1,c2,…ck,… be such that
 * $$\sum_{k=1}^\infty c_k^2$$
 * converges. Then there exists an element f ∈ R with c1,c2,…ck,… as its Fourier coefficients, i.e., such that
 * $$\sum_{k=1}^\infty c_k^2 = \lVert f \rVert^2$$
 * where
 * $$c_k = (f,\phi_k) \quad\quad (k = 1,2,3,\ldots).$$

 This suggests that the edit changing "converges uniformly" to merely "converges" is correct. --KSmrqT 07:02, 13 December 2005 (UTC)


 * Thank you very much. NatusRoma 08:05, 13 December 2005 (UTC)

Definitely "uniformly" is wrong. But please note that the anonymous editor did not change it to just plain "converges", but rather to "converges in L2". Michael Hardy 19:43, 14 December 2005 (UTC)
 * You're right. Before, it said, "converges uniformly in the space L2". I suppose that I should have given more context to the change when quoting it. NatusRoma 21:57, 14 December 2005 (UTC)

As of November 2008 (three years after the posts above), the article said converges normally, with a link to an article where this means that
 * $$\sum \|x_n\| < \infty$$

which is certainly wrong for orthogonal series in a Hilbert space. I made a change in the spirit of summable families, but finding an appropriate link would be better. Bdmy (talk) 21:07, 18 November 2008 (UTC)