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:

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Given a Euclidean space R, let {φ_k} 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,…c_k,… 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 {φ_k} in a complete Euclidean space R, let the numbers c₁,c₂,…c_k,… be such that

$\sum _{k=1}^{\infty }c_{k}^{2}$

converges. Then there exists an element f ∈ R with c₁,c₂,…c_k,… as its Fourier coefficients, i.e., such that

$\sum _{k=1}^{\infty }c_{k}^{2}=\lVert f\rVert ^{2}$

where

$c_{k}=(f,\phi _{k})\quad \quad (k=1,2,3,\ldots ).$

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This suggests that the edit changing "converges uniformly" to merely "converges" is correct. --KSmrq^T 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)