Talk:Orthonormal basis

Schauder Basis?

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)

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)

hello this talk is not good —Preceding unsigned comment added by 124.124.247.141 (talk) 06:13, 17 January 2011 (UTC)

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)

"an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal." A tautology is a phrase whose meaning is tautological. Good grief. 67.160.133.226 (talk) 16:35, 23 June 2013 (UTC)

Error

The article states: "An orthonormal basis of a vector space V makes no sense unless V is given an inner product; a Banach space does not have an orthonormal basis unless it is a Hilbert space, but in some a basis may exist if one does not demand it be orthonormal." This statement confuses an orthonormal basis with a regular (Hamel) basis. The axiom of choice guarantees that all vector spaces (normed or unnormed) have bases, so it's not true that only "some" Banach spaces may have basis, all of them do. What I believe it should say is "some Banach spaces have a Schauder basis which is similar to an orthonormal basis but without the orthogonoality condition".

The key difference of course being that a basis only allows finite sums while an orthonormal basis or Schauder basis allows infinite sums.203.167.251.186 (talk) 07:18, 25 January 2009 (UTC)

Question

Regarding their existence. The article states that every Hilbert space admits an orthogonal basis (by Zorn's Lemma) but that this will be uncountable if the space is not separable. My question is: for an uncountable orthonormal basis, do you need to define your sums over arbitrary indices or can you still get by using a countable index for your infinite sums? 203.167.251.186 (talk) 04:16, 4 February 2009 (UTC)

In a way, both answers are true. When you have an uncountable orthonormal basis $(e_{i})_{i\in I}$, you still have Bessel's inequality for every vector x,

$\sum _{i\in I}|\langle x,e_{i}\rangle |^{2}\leq \|x\|^{2}$

hence there is an at most countable subset J of I consisting of indices j such that the scalar product $\langle x,e_{j}\rangle$ is non zero (defining an arbitrary sum of non-negative reals is not a big deal, see Series (mathematics)#Summations over arbitrary index sets). You may thus say that

$x=\sum _{j\in J}\langle x,e_{j}\rangle \,e_{j},$

which can also be expressed as a more usual series if you enumerate the set J as $j_{0},j_{1},\ldots$. But this is not very pleasant nor elegant, because J depends upon x, and usually one prefers to give directly a meaning to the uncountable sum, see what follows in the same article on "series" linked above. --Bdmy (talk) 09:53, 4 February 2009 (UTC)

Thank you 203.97.49.105 (talk) 11:58, 6 February 2009 (UTC)

Here you are taking the closure rather than the sum, thus confusing two notions of basis. Tkuvho (talk) 09:00, 18 November 2010 (UTC)

Unitary matrices

I think this article should link to Unitary matrix, since that's what you get when you express an orthonormal basis as a matrix.

Existence?

The "existence" section currently says "Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits a basis and thus an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality. A Hilbert space is separable if and only if it admits a countable orthonormal basis."

Is there a source for these claims? First, our Gram-Schmidt process explicitly requires the basis to be finite. Furthermore, the comment appears to confuse two different notions of basis for a Hilbert space. Tkuvho (talk) 08:57, 18 November 2010 (UTC)

It's trivial to generalise the Gram-Schmidt process to transfinite bases using transfinite recursion. Are you the one who added "but not orthonormal base" to the "Existence" section? --Svennik (talk) 07:56, 29 January 2021 (UTC)

Too easy for an image?

Not for everyone, you know! — Preceding unsigned comment added by Koitus~nlwiki (talk • contribs) 23:26, 25 December 2018 (UTC)

*Not* every Hilbert space has an orthonormal basis? What?

> Using Zorn's lemma and the Gram–Schmidt process (or more simply well-ordering and transfinite recursion), one can show that every Hilbert space admits a basis, but not orthonormal base

Wait, what? "But not orthonormal base"? It's a well-known fact that every Hilbert space has an orthonormal base, which can be proven using (for instance) a straightforward generalisation of Gram-Schmidt. Also, what's the point of mentioning Gram-Schmidt if not to explain this fact? --Svennik (talk) 07:53, 29 January 2021 (UTC)

Expansion is incorrect?

The Basic Formula section states that

If $B$ is an orthogonal basis of $H,$ then every element $x\in H$ may be written as

$$x=\sum _{b\in B}{\frac {\langle b,x\rangle }{\lVert b\rVert ^{2}}}b.$$

When $B$ is orthonormal, this simplifies to

$$x=\sum _{b\in B}\langle b,x\rangle b$$

The inner product page defines the inner product as being linear in the first coordinate and conjugate linear in the second coordinate as the default. Using that convention, this formula has the inner product switched. For example taking the orthonormal basis $B=\{(1,0),(0,1)\}$ of $\mathbb {C} ^{2}$ with the standard inner product would give $(i,0)=\langle (1,0),(i,0)\rangle (1,0)+\langle (0,1),(i,0)\rangle (0,1)=-i(1,0)=(-i,0).$ With linearity in the first component, the correct formula is $$x=\sum _{b\in B}\langle x,b\rangle b.$$ Magidin (talk) 21:00, 30 January 2024 (UTC)

The error seems to have been introduced by Volker Weißman on May 24, 2021, and gone unnoticed. I will return the formula to the order consistent with the inner product page. Magidin (talk) 21:11, 30 January 2024 (UTC)