# Talk:Hilbert space

I'm an Econ PhD. I too think the article needs a simpler section. For example, I cannot tell if Euclidean space is a Hilbert space from the article as it currently stands. How about using Euclidean space as the first example? Show that it is a vector space, using the 8 criteria for that (inverse, addition, etc.)--- that is start  from the idea of a set, not of a vector space. Then show that dot product has the needed properties of multiplication.  Then show completeness (which is where I get lost for Euclidean space-- but others will have gotten lost earlier). editeur24 (talk) 16:55, 20 April 2023 (UTC)

Immediately after the lede, the very first section is titled Motivating example: Euclidean vector space. Did you give up before you got that far? Or is that section missing something? 67.198.37.16 (talk) 22:55, 25 November 2023 (UTC)

Can someone provide information about the self-duality of Hilbert spaces? 203.167.251.186 (talk) 06:56, 9 June 2010 (UTC)

See section "Duality", where the matter is discussed at length.  Sławomir Biały (talk) 10:29, 9 June 2010 (UTC)
Consult the book W.Rudin "Functional Analysis". The so-called dual Banach space H* of a Hilbert space H (space of all cont. lin. functionals on H) can be identified with H in that any F in H* can be written as F(x)=<y|x> for all x in H where y=y(F) is also in H. In that sense with conjugate linear identification H*->H, F->y(F), H becomes its own Banach space dual, i.e., self-dual.  — Preceding unsigned comment added by LMSchmitt (talk • contribs) 11:38, 3 November 2021 (UTC)

The lead says: "The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is a vector space equipped with an inner product, an operation that allows defining lengths and angles. Furthermore, Hilbert spaces are complete, "
Yet the page Euclidean space says
-- "there are Euclidean spaces of any nonnegative integer dimension,"
-- "[...] define a Euclidean space as a set of points on which acts a real vector space, the space of translations which is equipped with an inner product"
-- "With the Euclidean distance, every Euclidean space is a complete metric space."
So an important question for the current article to answer is what makes a Hilbert space (a term unfamiliar to many readers of this page) anything other than a Euclidean space (a topic familiar to a broader audience)?
And why is that difference important?
One difference might be that a Hilbert space can be over the complex numbers. But does that really do anything other than double the number of dimensions? And in any case, evidently there's already an extension of Euclidean spaces that includes complex dimensions: affine spaces.  Gwideman (talk) 02:58, 24 February 2021 (UTC)

The Hilbert space has (or potentially has, depending on definition) infinite dimensions.  "Any nonnegative integer dimension" of Euclidean space is not meant to include infinity. CyreJ (talk) 12:36, 17 March 2022 (UTC)

@David Eppstein: Greetings! I've noticed you've made an edit on this article. Does it still deserve its GA status? I thought it lacked inline citations, and was about to affix the template when I noticed the rating... Horsesizedduck (talk) 01:38, 17 July 2021 (UTC)

My recent edit was quite minor. I agree, though, this looks under-referenced. Other than that it appears from a superficial look to be in pretty good shape, though. It needs effort to bring it back into GA shape but I think it should be possible. —David Eppstein (talk) 01:42, 17 July 2021 (UTC)
More careful reading shows that the sourcing is not as bad as it looks. Some sections have catch-all footnotes near the front like "See this text for this entire section, but really you can find all this in any good textbook on the subject", and these footnotes are correct. This is basic textbook material that doesn't need close sentence-by-sentence sourcing for adequate coverage. —David Eppstein (talk) 05:40, 17 July 2021 (UTC)

LMSchmitt has edited the first paragraph of the lead with the edit summary "Improved some English (which was BAD). If one talks about "complete", then one needs to name the DISTANCE d, in this case, one needs to name that d comes from the inner product. This was incoherent before. Bad Math." In fact, the edit consisted mainly in introducing an unnedeeded symbol H for the Hilbert space, and unneeded WP:JARGON and technical terms (canonical, complete metric space, Banach space). The edit did not fix what it was supposed to fix, namely, that it must be clear that the distance is defined from the inner product. So, I have reverted the edit. I was immediately reverted with an edit summary consisting essentially  in irrelevant personal attacks. So, I have reverted again, with an edit summary containing links the the guidelines that apply here. By the way, I have fixed several other impropernesses of the formulation that I did not remark before. 
So, if LMSchmitt disagree with the current formulation, they must apply WP:BRD, that is discuss the first paragraph here, and not edit warring. D.Lazard (talk) 14:43, 15 September 2021 (UTC)

old version based upon a reading error removed. New almost identical version addad below — Preceding unsigned comment added by LMSchmitt (talk • contribs) 06:07, 16 September 2021 (UTC)
Please, do not forget to sign your posts in talk page with four tildes (~~~~). This would avoid to other editors the boring task of doing it for you. I have also indented your post as recommended in Help:Talk pages#Indentation. D.Lazard (talk) 07:01, 16 September 2021 (UTC)
Sorry, I forgot to sign. LMSchmitt 07:53, 16 September 2021 (UTC)
D.Lazard has previously removed contributions of other editors which were language-wise in order and also content-wise in order without being able to give a solid reason why he did so. [Upon request, he kindly gave an explanation of his reason which however didn’t make sense.] This can be documented within WP and was verified with native speakers and mathematical experts. Also in the present dispute, he first reverted my edit, almost without acknowledging its merits. While avoiding edit warring by himself, he actually incorporated in his 2nd revert-edit almost all of the concerns which I tried to fix with my initial edit of the first paragraph of the present article. Now, he takes credit for fixing these things.
Things done in my first edit: (more than D.Lazard claims)
Removal of the article <a> in a sentence. [old]. In mathematics, a Hilbert space (named for David Hilbert) generalizes the notion of Euclidean space. [new]. In mathematics, the term Hilbert space (named for David Hilbert) generalizes the notion of Euclidean space. [--] Speaking of <a Hilbert space> is speaking of a mathematical object (Ex.: Incidentally, a Hilbert space is complete.). Not a notion.  So, the notion of (or <the term>) Hilbert space generalizes the notion of Euclidean space. There are earlier versions of this article which seem to concur with this. I found the old version bad English using the article <a> which also could simply be dropped.
A Hilbert space is no methods (or meta-methods) which can be extended. [old]. It extends the methods of linear algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. [new]. It allows to extend the methods of linear algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to vector spaces of any finite or infinite number of dimensions. [old] is simply false. A Hilbert space is a well-defined mathematical structure. It does not extend the methods of linear algebra and calculus rather these methods can also be applied to it. [new] is correct. Given the structure of Hilbert space, one can apply/use/extend many methods from 2,3D linear algebra and calculus to obtain interesting results such as results about the distance of a point to a convex set. This, essentially my wording using <allow>, is in D.Lazard's 2nd revert-edit. Why a partly delete to something understandably worse first.?
D.Lazard has acknowledged with his first revert-edit that a previous serious FLAW in the first paragraph of this article was that there was no connection made between completeness and the metric that comes with the inner product. I pointed that out in the header/summary of my first edit as he quotes above. Instead of perceiving the merits of my contribution, D.Lazard  resorts to <value-bashing>, i.e., wording whose value can be seen differently is disqualified as jargon etc. If one speaks of <canonical distance>, then this means in English while talking math (Oxford dic) that one speaks of a distance which arises naturally in a given context (according to a known formula). With that wording <canonical distance> used after it was mentioned that the inner product allows for a distance to be defined, the connection between inner product (which defines a canonical distance as in 3D space) and completeness was definitely made. D.Lazard's statement
The edit did not fix what it was supposed to fix, namely, that it must be clear that the distance is defined from the inner product.
is blatantly false for the purpose of being right after the fact. Had he understood my first edit, he wouldn't have simply deleted my version, but built on my editing in a constructive way.
Introducing a letter H did simplify a sentence significantly. In my opinion, it was useful and the simplification it caused made it necessary.
The old article itself pointed to "complete metric space" as the current version does. So why bashing me.?
A Hilbert space is a Banach space and that important fact should be mentioned at the top of any article dealing with it, as it allows Banach space techniques to be applied such as duality and weak compactness.
D.Lazard removed my contribution in a haste without taking its merits into account as is obvious from the many elements of concern that I have put forward and that he incorporated in his 2 revert-edits. He takes credit for fixing points that I have raised and had already corrected as documented above. LMSchmitt 08:13, 16 September 2021 (UTC)
Here are some flaws of the old version that are badly corrected in LMSchmitt's version, and are now fixed:
A mathematical structure (old version) or a term (LMSchmitt's version) cannot generalize a notion. Now formulated as "Hilbert spaces allow generalizing ..."
It is wrong, or at least confusing, to say (both versions) that Hilbert spaces generalize Euclidean spaces, as many important properties of Euclidean spaces cannot be extended to Hilbert spaces. Now fixed with the formulation "Hilbert spaces allow generalizing the methods of linear algebra and calculus from ... to ...", which fixes also the second concern of LMSchmitt.
Distance: I agree with LMSchmitt that when using "distance", it must be clear between what the distance must be considered. But LMSchmitt solved this by complicating the wording, introducing a symbol for naming the Hilbert space, and calling "vectors" its elements. Using vector here is not convenient, as, in many Hilbert spaces, typically in functional analysis, the elements are not called vectors, but functions. I have solved this by saying that the inner product "allows defining a distance function". This would be considered as too vague in the body of the article, but this seems clear enough in this paragraph, while being mathematically correct.
Angles: AFIK, "angles" are generally not considered in Hilbert spaces, except for right angles. Therefore, I have replaced "angles" by "perpendicularity (known as orthogonality in this context)". "Perpendicularity" is here because I believe that,  for many readers, it is less technical than "orthogonality". If I am wrong, "Perpendicularity" could be removed.
Technical words: Without a definition, "canonical distance" is mathematical jargon, and must be awoided. "Complete" is a part of the definition of a Hilbert space. So it must appear here with a link. The fact that the best link is to Complete metric space does not implies that the phrase "complete metric space" deserves to be displayed. Similarly, Banach spaces are an important generalization of Hilbert spaces, but mentioning them in the first paragraph is unhelpful for readers that know them and confusing for others.
So, the current version fixes some issues that were pointed out by LMSchmitt, and some issues that they did not considered. It fixes also issues introduced by LMSchmitt's edit. So, let other editors validate or challenge these assertions. D.Lazard (talk) 08:56, 16 September 2021 (UTC)

A Hilbert space H has a norm coming from the inner product by definition. The norm must be definite, i.e., ||x||=0 iff x=0 also by definition. The one-point set S={0} c IR is closed in IR, hence Lebesque measurable, and the characteristic function of S (say chi_S) is a positive Lebesque-measurable function which, thus, can be Lebesque-integrated over IR which yields 0. One has that chi_S is a so-called Null-function, i.e., it's a function that has support on a Lebesque-measure 0 set. In order to make the space L2 of sq-integrable functions a Hilbert space, one has to consider L2/N where one factors out the space N of Null-functions. This annoying, minor inconvenience is often "suppressed", i.e., everything is considered mod N really. ---- This should be mentioned, since without this modification (or philosophy of ignoring N), L2 is not a Hilbert space since it has no definite norm.  — Preceding unsigned comment added by LMSchmitt (talk • contribs) 12:00, 3 November 2021 (UTC) 

||f||=0 (norm induced by inner product) iff f=0 a. e., so L^2 is a Hilbert Space. (it is also complete wrt the norm)  — Preceding unsigned comment added by 2604:3D09:797D:3500:2D1B:83C2:EC44:75D (talk) 02:44, 17 March 2022 (UTC)

David Eppstein, following on from the recent discussion at WT:GAN, where this article was quite literally top of the list of potential GAR reassessments; I can see how some paragraphs are split by mathematical symbols, but at the same time you have entirely unsourced sections of pure prose (Probability theory, Color perception) and sections with mathematical lines which don't have any citations anyway (Pythagorean identity, Bounded operators, etc.) I appreciate that this is all "standard stuff" in textbooks, but if you could cite the standard textbooks that would be very helpful in not having to bring this to GAR. ~~ AirshipJungleman29 (talk) 10:54, 7 August 2023 (UTC)

I agree that the sections you list are unsourced and need sources. But my past contributions to this article have been quite minor; it hasn't even been on my watchlist. I could probably handle some of these, and I'll add it to my (long) list of parts of Wikipedia to work on, but it would take quite a bit of time, maybe a month, if I'm doing it all myself. Perhaps more active recent contributors such as User:Tito Omburo or User:D.Lazard could pitch in.
For those contributors: The context is that Wikipedia's Good Article process recently made its sourcing rules more strict, requiring all text that is not a summary of later sourced content to have an inline footnote, no later than the end of the same paragraph. User:AirshipJungleman29 is very keen to apply these standards retroactively and immediately to all past GAs as well, setting up a big cleanup effort to turn content formerly evaluated as good into sourced content or (possibly more likely) to decimate our listing of old Good Articles. The only reward for doing this cleanup is that you avoid getting Good Article credit getting taken away from some other long-past contributor. It doesn't help most readers much because the prose is generally in good shape, just badly sourced. A lot of the most problematic articles are in mathematics and there are very few active Good Article contributors in mathematics (mostly me although there are some others who have the expertise but usually contribute elsewhere). —David Eppstein (talk) 16:26, 7 August 2023 (UTC)
Thanks for the summary, David; I'd like to add that the reason I'm not hanging about waiting for sections to be cited is that, as noted above, there are very few active contributors, and so any content which currently doesn't meet GA standards is likely to remain sub-quality for the next decade or more. Also, there's no prejudice against you taking GA credit for the article—no one's checking! ~~ AirshipJungleman29 (talk) 16:35, 7 August 2023 (UTC)
"sub-quality" seems like a largely arbitrary summary. Text that is easily verifiable and even has relevant sources linked at the bottom but doesn't have a footnote littered after every sentence is not really notably worse in quality or less useful to readers than text which does have such footnotes. Maybe "sub quality" could be defined as "spent insufficient effort checking bureaucratic tick boxes". –jacobolus (t) 08:25, 8 August 2023 (UTC)
Probably—after all, the entirety of Wikipedia is a bureaucratic exercise in ticking boxes. Then again, I don't see how such text is verifiable if it doesn't have a footnote after it, and just gestures vaguely at a mass of sources. Many thanks to you and Tito Omburo for making a start, though. ~~ AirshipJungleman29 (talk) 08:47, 8 August 2023 (UTC)
A footnote doesn't itself make text verifiable. A reader still has to understand the given source well enough to see that it does, in fact, verify the text it's supposed to. For an article like this, verifying any of the cited claims will require at least a year or two of university-level mathematics background. Consequently, a reader who can use the citations at all won't need them after every sentence, and probably not for every paragraph. What's the point of footnoting each paragraph in a section with "Chapter 12 of Smith (1980)", "Chapter 12 of Smith (1980)", "Still in chapter 12 of Smith (1980)"...? XOR'easter (talk) 16:30, 8 August 2023 (UTC)
entirety of Wikipedia is a bureaucratic exercise in ticking boxes – If you really think this perhaps it would be better to find something more useful to do with your time. I would say that the vast majority of effort spent on Wikipedia is research and writing with the goal of conveying meaningful explanations to readers, and discussions (and sometimes conflict resolution) associated therewith. But to answer your concrete question: to verify the material here a reader would look at the linked sources, and turn to the relevant chapter or section of the source, and skim down to where the topic is addressed. This is not made significantly easier by a footnote for every line (though including one (1) hyperlink in each section of the article to the relevant chapter of a well written online textbook or scan of a paper textbook would be helpful, by saving the reader the trouble of checking a book out from the library). In the case of this article in particular (and many other technical articles covering basic parts of technical curricula) there are hundreds of sources containing more or less the same material; if you don't like the sources specifically linked, just pick up any textbook about the topic. –jacobolus (t) 17:59, 8 August 2023 (UTC)
I agree that the sourcing could be better in places.  I've tried to add sources any obvious place I could find.  As others have observed, most of this content can be easily sourced to general textbooks.  In fact, the biggest problem I have had is choosing from a number of textbooks which one is the best to cite.  If there are any other places that anyone thinks are insufficiently sourced, let me know.  Tito Omburo (talk) 12:22, 8 August 2023 (UTC)
That's a very nice problem to have. As for other places where an inline citation would be nice:
The last paragraph of the Sturm–Liouville theory section
The first paragraph or two of the Ergodic theory section
The first and last paragraphs of the Fourier analysis section
The first and third paragraphs of the Probability theory section
The last paragraph of the Duality section
A couple for the latter half of the Bounded operators section
Similarly for the Direct sums and the Bessel's inequality and Parseval's formula sections
So a dozen or so citations in total, Tito Omburo. Probably shouldn't be a problem, if you don't awfully mind. ~~ AirshipJungleman29 (talk) 15:35, 8 August 2023 (UTC)
There's only one paragraph in the Sturm–Liouville theory section (and it already has a footnote). XOR'easter (talk) 16:22, 8 August 2023 (UTC)
So it does, XOR'easter, but it looks like someone's added another, so that's nice of them. ~~ AirshipJungleman29 (talk) 10:58, 10 August 2023 (UTC)

In the introduction, I changed "an inner product that defines a distance function" to "an inner product that induces a distance function". "Define" is technically correct, but the connotations are off. A definition is something that a human imposes, whereas the relevant metric is an automatic consequence of the definition of an inner product, and I feel that it's more accurate to say therefore that the latter "induces" the former. It's a small thing, I know, but I wanted to set out my reasons properly, and the edit comment seemed like a bad place for all this detail. —Calisthenis(Talk) 17:56, 16 August 2023 (UTC)

The redirect Hilbert spaces and Fourier analysis has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 March 28 § Hilbert spaces and Fourier analysis until a consensus is reached.  1234qwer1234qwer4 01:32, 28 March 2024 (UTC)

There is a claim in the "Hilbert dimension" subsection that "The Hilbert dimension is not greater than the Hamel dimension (the usual dimension of a vector space). The two dimensions are equal if and only if one of them is finite." The "only if" in the second sentence is wrong. The two dimensions are certainly equal when the space is finite-dimensional, but they can be equal even when the space is infinite-dimensional. For example, the sequence space l2 on the continuum has both Hilbert dimension and Hamel dimension being the continuum. More generally, this answer on Math StackExchange shows the sequence space l2 on any set of cardinality 2א‎α for any ordinal α has its Hilbert dimension equaling its Hamel dimension (namely, both are 2א‎α).
I think if the "if" direction of the statement is kept, then an additional sentence noting that the two dimensions can still be equal when the space is infinite-dimensional should be added. In that case the Math StackExchange answer might need to be cited. I'm new to Wikipedia editing, so I'm not sure how to do that. (Also, an example that the two dimensions can be different might need to be added as well. The standard example is just the sequence space l2 on ω, which has Hilbert dimension א‎0 by definition and has Hamel dimension continuum, which is a consequence of the main result of this paper.) David-Gao-200008 (talk) 08:29, 29 January 2025 (UTC)

Thanks, I have deleted the incorrect statement. Tito Omburo (talk) 11:54, 29 January 2025 (UTC)