Talk:Orthonormal basis

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Question: How does an orthonormal basis differ from a Schauder basis? --

Please, 'Orthonormal basis' is a general concept, not specific to Hilbert space. -- looxix 00:58 Mar 12, 2003 (UTC)

It is specific to inner product spaces. Hilbert spaces are complete inner product spaces. How much ink is spilled on inner product spaces that are not complete, except when writing also about spaces that are complete? Michael Hardy 20:19 Mar 12, 2003 (UTC)

Several mathematical errors were introduced into this article today. I have corrected them. Note:

  • It is impossible to define an orthonormal basis on any vector space unless it has an inner product.
  • Hilbert spaces and inner product spaces are not more general than vector spaces; rather, vector spaces are more general than Hilbert spaces or other inner product spaces.
  • An orthonormal basis is not generally a "basis", i.e., it is not generally possible to write every member of the vector space as a linear combination of finitely many members of an orthonormal basis. And as soon as you're using infinitely many, then the question of which kind of convergence you're talking about arises: almost-everywhere convergence, L_2 convergence, etc. (Those two are perhaps the most important in this case.)

Michael Hardy 19:22 Apr 14, 2003 (UTC)

Sorry for that. But the article as it stands might as well be gibberish for the non-expert reader. It needs an opening which explains in non-technical terms what one is, why it is important, what it is used for, etc. -- Tarquin 22:58 Apr 14, 2003 (UTC)

<mood> The definition is a standard part of the undergraduate-level mathematics curriculum; usually I construe "non-expert" as meaning a person with a PhD in mathematics who does not specialize in a particular research area. How about if we compromise and say "incomprehensible to non-mathematicians" (although that seems a bit exaggerated)? </mood> Michael Hardy 01:56 Apr 15, 2003 (UTC)

I've given this article some attention and format work. There is still a duplicated discusion about 'basis' - where should this be placed?

Charles Matthews 09:26, 11 May 2004 (UTC)

[edit] Merge?

I just ran across the Orthonormal function system, which might be better placed in this article. But I feel it currently says the wrong thing. Usually when I hear the phrase "system" I don't necessarily expect it to be a basis. I think the best solution might be a short article about orthonormal systems that links here. Thenub314 (talk) 01:40, 27 April 2008 (UTC)

[edit] Simplifying the intro

I simplified the first sentence by moving to next sentences the specific info about infinite-dimensional Hilbert spaces, which in my opinion is secondary for most readers (although it is important that it is provided in the last part of the introduction). I am not familiar with infinite-dimensional spaces, but I carefully respected the information given in the previous version. I believe that the result is much clearer and better organized than the previous version. Please let me know if you agree. Paolo.dL (talk) 18:51, 2 June 2008 (UTC)