Talk:Hilbert space

Simpler section

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)

Reconciling Hilbert space with Euclidean space

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)

GAR

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)

"Define" vs "induce"

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)

"Hilbert spaces and Fourier analysis" listed at Redirects for discussion

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)

The condition for when Hilbert dimension and Hamel dimension are equal is wrong

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 2^א‎_α for any ordinal α has its Hilbert dimension equaling its Hamel dimension (namely, both are 2^א‎_α).

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)

mathbb

The article in various places uses mathbb (inside the < math > tags), and in other places, standard bold, for the real and complex fields. I think the article should stick to one convention or another. I don't have any preference. Best if someone just changes everything to whatever convention works best with the current Wikipedia style regime. Sławomir Biały (talk) 08:42, 9 February 2026 (UTC)

I switched them all to ordinary bold, which I think was the original convention here. –jacobolus (t) 16:52, 9 February 2026 (UTC)

Article review

AirshipJungleman29 Opened a discussion on this article's GA status in 2023, but since that was a while ago and there have been other discussions, I decided to start fresh. This is one of the last articles listed at WP:SWEEPS2023.

I went through the article and added citation needed tags to places where I thought they were needed. I also noticed that the article is quite big, at over 10,000 words, and should probably have sections spun out or summarised more effectively. Are editors interested in resolving concerns, or should this go to WP:GAR? Pinging other discussion participants @Jacobolus, Tito Omburo, and XOR'easter: Z1720 (talk) 20:44, 28 March 2026 (UTC)

Many of these citations are easily resolved either by citations already at the beginning of a paragraph, or even citations that already appear in the sentence that the tag was applied to. Some others summarize standard facts from other articles (for which citations can easily be supplied). The few others not in these categories are standard enough that they should be citable to standard reference sources. A GAR is not required. Sławomir Biały (talk) 20:51, 28 March 2026 (UTC)

Thanks for the ping, but I'm currently focusing not on review but on improvement, so I won't be participating here further. ~~ AirshipJungleman29 (talk) 22:02, 28 March 2026 (UTC)

• @Sławomir Biały: Thanks for working on this article. The article is still quite large and I think information can be spun out or summarised more effectively. Could you or other talk page watchers take a look and see if any information can be moved or trimmed? Thanks, Z1720 (talk) 02:02, 4 May 2026 (UTC)

  Why do you think that? Do you have some specific part that you think is off topic, unimportant, excessively tedious, etc.? You can't judge whether an article's content is useful and relevant with just a word count. –jacobolus (t) 02:13, 4 May 2026 (UTC)

• @Jacobolus: I think lots of the information in "Applications" can be spun out into their particular articles, where the ideas can be described in more detail while this article summarises that information. In particular, "Fourier analysis" and "Probability theory" are where I would begin this work because they seem the most bloated. "Definition" has two examples: are they both needed? I think the "History" section be copyedited for excessive words and detail: I try to have 2-4 paragraphs per section/subsection, and this has approximately six. I think if the article underwent a copyedit, other excessive/redundant words or duplicate examples could be removed or spun out. Z1720 (talk) 02:53, 4 May 2026 (UTC)

  Why would you expect to trim off the history section? The article describes the original background to a layperson in a simple way, recording another development and generalizing the whole thing.

  If you compare Prime number, which has a little difference between this article and Hilbert space, its history section mentions the mathematicians who discovered the connection to such numbers, particularly those who managed to use computers and ongoing research. The subsection then answers the layperson's questions on what happens if 1 is considered a prime number.

  I could argue similarly on the application sections for both articles. They are widely useful to apply to other fields if you are using Hilbert space and Prime numbers. And this already meets the criteria on broad coverage. Dedhert.Jr (talk) 04:27, 4 May 2026 (UTC)

  Arguably there could easily be more history, and also several additional applications, each discussed at greater length. Hilbert spaces are a fundamental topic. Why do you think it's "excessive", and why do sections need a precise paragraph count? –jacobolus (t) 05:25, 4 May 2026 (UTC)

• I agree with jacobolus. The history section is fine length-wise, and even could be expanded. I think word-count is a poor basis for determining whether content can be properly summarized in less space. The main issue would be breadth in coverage, and the article I would say mostly meets that: there is very little I would add here. There is also very little I could see justifiably cutting. I hadn't read the article in some time, probably since the original GAN, so I was afraid I might find that it had drifted, but it has stayed pretty tight I think. My basic assessment of some of the sections:

  The Spectral theory section is good, long but summarizes the literature and genuinely useful to readers.

  The Constructions section is fine, although perhaps a different section heading could be devised.

  Orthonormal bases is mostly fine, but I would cut the Quantum field theory section or merge a shortened version with the earlier quantum mechanics applications section. Orthogonal complements, also ok.

  The applications sections seems appropriately summative. I believe the only significant expansion of the article from my last review is the long Probability section in applications (an important application). The length of that section is mostly earned, because it describes the Hilbert space projection interpretation of conditional expectation and martingales, which certainly belongs here. I am not so sure about the paragraph on Ito calculus; I would not delete it, but might try to make it segue into the last paragraph more directly. The last paragraph of the probability section genuinely is important for the article, since the Cameron-Martin space is probably the most important direct application of Hilbert spaces to probability theory.

  To my eye, the only section that looks like it needs major work (technical cleanup) is the Operators section, but the result is likely to make it longer, not shorter.

  Regarding the two examples in the Definitions section, I could potentially see the example of sequence space being moved to the Examples section, but it is a necessary example for the article to cover somewhere. The two examples are obviously rather different: one is motivation, being only finite dimensions; the other is an example of an "honest" Hilbert space (and is in a precise sense the universal separable Hilbert space). I did not see any "duplicate" examples.

  I didn't find a lot of redundancies. There are several places where the same idea is used in different contexts (e.g., the spectral theorem is repeatedly invoked, orthogonal projections are repeatedly referred to), but these are reflections of the fact that some ideas in Hilbert space theory are needed in different places in applications, not that a big consolidation of everything that mentions the spectral theorem should be organized under "Spectral theory", etc. Sławomir Biały (talk) 09:00, 4 May 2026 (UTC)

• The 8,000 guideline was discussed at WT:AS, and didn't lead to consensus to change the numbers (some want more, some want less, some thought keeping it at 8,000 was fine). Right now, the 8,000 is a guideline: not a hard-and-fast rule, but editors should consider spinning out when it exceeds that word count. If the History section should be expanded, as suggested above, I would recommend spinning out that section into a new article and summarising the information here: since the article is already so large it would be better to bring information to other articles (where it can be described in more detail) and the most important aspects can be highlighted here. I also added citation needed templates (and a "when?" tag) to places that might need citations or to have the citation moved to the end of the paragraph. Z1720 (talk) 14:41, 4 May 2026 (UTC)

  If it were spun into a new expanded article, the material currently in this article could be left as a summary. It would not be helpful to significantly shorten it here. –jacobolus (t) 14:47, 4 May 2026 (UTC)

  I don't really understand why you seem to think the History section is too long. Perhaps you favor deleting the history section altogether? Sławomir Biały (talk) 15:05, 4 May 2026 (UTC)

• @Sławomir Biały: No, I do not want the "History" section removed from the article. I think it can be summarised more effectively. If there is more information to add to that section, as implied above, then it might be time to create a "History of Hilbert space" article (if the history article would be notable on its own). Z1720 (talk) 16:28, 4 May 2026 (UTC)

• What, in your view, should be deleted from the existing section? Sławomir Biały (talk) 16:29, 4 May 2026 (UTC)

Here are some sections of which could potentially be removed, as it seems like additional information that is not necessary to understand the overall history of the term and its use:

• "Functions can be added together or multiplied by constant scalars, and these operations obey the algebraic laws satisfied by addition and scalar multiplication of spatial vectors."
• "Von Neumann used them in his seminal work on the foundations of quantum mechanics,[35] and in his continued work with Eugene Wigner." Von Neumann is already mentioned as coining the term "abstract Hilbert space", and the history section doesn't need to list everytime it is used in significant mathematician's research.
• "In short, the states of a quantum mechanical system are vectors in a certain Hilbert space, the observables are hermitian operators on that space, the symmetries of the system are unitary operators, and measurements are orthogonal projections." This sentence is an explanation of the mathematical concept, and not the historical significance. It can be explained in another article or moved to an appropriate section in this article, if needed.
• "The algebra of observables in quantum mechanics is naturally an algebra of operators defined on a Hilbert space, according to Werner Heisenberg's matrix mechanics formulation of quantum theory." I am not sure what this has to do with the history and can be trimmed.
• "These techniques are now basic in abstract harmonic analysis and representation theory." I don't think this is necessary for this section and can be removed.

While reading through the section I did some trimming of excess words: I invite editors to take a look and make necessary corrections or changes. These are suggestions above, and some of them might be needed, and other editors can WP:BEBOLD and try to trim or summarise other information more effectively. Unfortunately, I don't have time or the will to go through the entire article to do this, and suggest that editors who are interested in this article keeping its GA status take a look themselves. In short, if those editors read through and are not able to trim very much, I recommend that a second read through happen where a stricter criteria is used to decide what can stay and what can be spun out. Z1720 (talk) 16:58, 4 May 2026 (UTC)

To be frank, most of your suggested changes seem harmful. I feel like you have a (misguided) idea that "history" and "technical detail" are incompatible and must be separated. But when discussing the history of a technical topic, doing so completely obscures the point. –jacobolus (t) 17:58, 4 May 2026 (UTC)

@Jacobolus: Technical explanations are sometimes not necessary and can create cluttered and disorganised prose. Articles that are larger in size make it more difficult for the reader to find specific information they may be looking for, negatively affect load times (especially for users on less reliable internet connections) and increase the difficulty of maintaining the article. I have given suggestions on what could be moved/removed, but other suggestions are also welcome. However, I do not think the article will adhere to WP:GA? 3b if it maintains its current size. Z1720 (talk) 18:11, 4 May 2026 (UTC)

Unfortunately, I don't have time or the will to go through the entire article to do this

I think the problem seems rather that you don't have the technical expertise or time/effort to make sense of the claims being made, so your proposed changes are all completely superficial and indifferent to the content. Instead of trying to make the explanation as clear as possible for readers, you are trying to tick some boxes on a checklist somewhere (for that matter, using a made up personal interpretation of the checklist items which is not reflective of the checklist text itself). –jacobolus (t) 18:12, 4 May 2026 (UTC)

@Jacobolus: The checklist I use in my reviews is the GA criteria. If you think there is an aspect of the criteria that I am applying incorrectly, I encourage you to get more opinions on the topic at WT:GA or other appropriate noticeboards. I am happy to have any aspect of the criteria clarified to ensure that my reviews are the best quality possible. Z1720 (talk) 18:22, 4 May 2026 (UTC)

You have replaced the text "without going into unnecessary detail" (which is intentionally vague so that it can be adapted to match the context – what counts as "necessary" or not is a matter of local consensus, ideally based on nuanced expert opinions) with your own interpretation "does not contain X number of paragraphs / Y number of sections / Z total word count", an artificial constraint which is, in my opinion, extremely harmful to mindlessly pursue. –jacobolus (t) 18:58, 4 May 2026 (UTC)

@Jacobolus. Agree. The review is personally like you have to follow guidelines according to the number of bytes, rather than focusing on the quality of clarifying technical things to laypersons. If some topics in the history section are trimmed off, this might violate another GACR on broad coverage and technical problems. Dedhert.Jr (talk) 01:07, 5 May 2026 (UTC)

All of those are sentences that I would keep. Three of the items directly address one of the most important applications of Hilbert spaces, indeed one of the deepest truths about physical reality that science has yet discovered, i.e., how quantum mechanics works. Stepwise Continuous Dysfunction (talk) 16:35, 6 May 2026 (UTC)

"Induced"

The lede contains the words "...a complete metric space with respect to the metric induced by the inner product". What is the meaning of the word "induced" here? Surely it's the topology that is induced by the metric, not the metric by the inner product? I think the word "defined" might be better here instead of "induced". Or am I misunderstanding the meaning? — The Anome (talk) 10:15, 21 May 2026 (UTC)

Defined is the intended meaning, but induced seems unproblematic to me. Sławomir Biały (talk) 10:31, 21 May 2026 (UTC)

Is it really even "defined"? There is not a separate new definition for each inner product. "Derived from the inner product" maybe? —David Eppstein (talk) 15:34, 21 May 2026 (UTC)

First sentence

Should the first sentence of the article be In mathematics, a Hilbert space is a complete inner product space.? Sławomir Biały (talk) 15:45, 22 May 2026 (UTC)

• No. Clearcut violiation of MOS:FIRST: "The first sentence should introduce the topic, and tell the nonspecialist reader what or who the subject is, and often when or where. It should be in plain English." Contrary to this edit summary, both the inner product and completeness are already introduced in the first paragraph. Normally, I would say it makes sense to add "A Hilbert space is formally defined as a complete inner product space", after the accessible lead paragraph. But it is not actually necessary here: the lead already does define the subject exactly to anyone reading the first paragraph. Sławomir Biały (talk) 15:45, 22 May 2026 (UTC)

No. I agree that such phrasing is far more technical than is necessary or desirable. To add to the comment above, there are many potential readers (like undergraduate students of physics) who are likely to see the term "Hilbert space" before they are familiar with the idea of a complete metric space. Indeed, they probably learn what completeness of a metric space means as part of learning what Hilbert spaces are. No one benefits from defining the familiar in terms of the obscure. Stepwise Continuous Dysfunction (talk) 16:10, 22 May 2026 (UTC)

Comment. Note that the point is not the sentence that you mention, but the problem with the current lede paragraph. The current lede paragraph does not define the subject properly since it mere says that Hilbert spaces are required to be complete. 慈居 (talk) 16:17, 22 May 2026 (UTC)

I have added the precise definition at the end of the first paragraph. By the way, I don't think "over the reals or complex numbers" in the previous stable version was necessary, because Hilbert spaces are also defined over any fields having (suitable notion of) norms. Also, the inner product space condition already has some restriction on what base fields they can have. 慈居 (talk) 16:32, 22 May 2026 (UTC)

I do not think that adding the precise definition at the end of the opening paragraph clarified things, at least not with that exact phrasing. As written, it looked like the last line was just repeating what had gone before. Some more careful phrasing would be necessary to indicate that the definition of "Hilbert space" includes exactly those conditions and no others. And I am not sure that the opening paragraph needs to do that anyway. The fraction of the reading population who are curious if there are any other details or qualifications will have their curiosity satisfied by the formal definition section below. Stepwise Continuous Dysfunction (talk) 16:57, 22 May 2026 (UTC)

Many of the readers (including me) use this article just for a quick look-up. It is annoying to be pointed to the middle of the article to look up the definition of a concept having a one-sentence definition. 慈居 (talk) 17:13, 22 May 2026 (UTC)

That annoyance, while understandable, is not a consideration in our editorial guidelines. The lead section, and especially the first paragraph, needs to be readable by as wide an audience as possible. Articles, and especially leads, aren't written for experts. They are written for general readers. Sławomir Biały (talk) 17:17, 22 May 2026 (UTC)

It is ironic to exclude a considerable group of readers by the name of "general" readers, isn't it? Anyway, I still hope a brief definition can be added in the lede in one way or another, yet I do not intend to challenge the deeply seated tradition of English Wikipedia community, as long as it does not affect its sister Wikimedia projects. Actually I figured out what to do that is satisfying to me. Please feel free to revert my edits and discuss. 慈居 (talk) 17:57, 22 May 2026 (UTC)

There is a true problem with this article, but inserting the technical difinition in the lead seems not a right solution. The problem is that the article is structured as a chapter of a textbook where the motivation of introducing a concept may precede the definition. Here, the motivation for reading the article may depend on the reader. So, it seems better to have first a section § Definition, then a section explaining why this definition allows unifying the treatment of seemingly different poblems, and then sections on the two prototypical examples (Euclidean spaces and absolutely convergent sequences). Also, a lot of trivia must be removed, since, for example, I cannot imagine how a reader who ignores what is a vector space or a Euclidean space could be interested by this article. D.Lazard (talk) 18:10, 22 May 2026 (UTC)

I don't really agree with this. The motivating example is not decorative: it explains what an inner product is, what completeness is, and how complex numbers come in. These are essential for someone to even begin to understand the definition. (It certainly isn't trivia, but maybe I misunderstand what you mean by that word.) That is to say, the first "prototypical example" is not intended merely as an example of a Hilbert space, but a familiar framing where the ideas necessary for understanding the definition are presented. The sequence space, likewise, serves an important function (and again, is not trivia: this content would have to appear somewhere in the article, even if not in its exact location). Specifically, it shows exactly how infinite dimensions come in, and how completeness enters in the infinite dimensional case. In other words, the examples are not just illustrative, but serve as necessary explanations of the concepts involved. Sławomir Biały (talk) 18:24, 22 May 2026 (UTC)

By the way, I doubt Euclidean space is a "motivating" example. Perhaps L2 spaces are better motivation. For Euclidean spaces linear algebra is enough. 慈居 (talk) 20:30, 22 May 2026 (UTC)

A Euclidean space is literally a finite-dimensional Hilbert space. More importantly, as I explained above, it is not merely decorative motivation: it introduces the ideas needed to understand the definition in a familiar setting. I do not think further repetition of this point will be useful, so I am going to leave it there. Sławomir Biały (talk) 20:39, 22 May 2026 (UTC)

I don't think you get my point. The historical motivation for Hilbert spaces (and the functional analysis) is more likely infinite-dimensional spaces of functions that arose in the practice. 慈居 (talk) 21:00, 22 May 2026 (UTC)

Ideas can be posed in a diffent way and I believe preliminaries are better given by links or appendices. 慈居 (talk) 21:02, 22 May 2026 (UTC)

This is a matter of taste, I suppose, but in the present context I don't think that links or appendices would be the best solution. That would involve (what seems to me like) too much flipping to other articles or to the end of this one in order to cover something that naturally arises early in the treatment of the topic.

Moreover, I don't think that the original historical motivation is necessarily the best conceptual motivation for an introductory overview. Stepwise Continuous Dysfunction (talk) 23:49, 22 May 2026 (UTC)

The historical motivation for Hilbert spaces was to apply the intuition and familiar geometrical tools people had developed in the context of finite-dimensional Euclidean vector spaces to the new context of infinite-dimensional function spaces. So briefly explaining that conceptual context is essential for conveying the historical development and current use of the idea. –jacobolus (t) 00:45, 23 May 2026 (UTC)

A problem with such a section is that you are more or less forced to read it because you don't know what conventions unexplained elsewhere the rest of the article is going to use. And any edits to make the definition section self-contained is likely to be considered redundant. 慈居 (talk) 18:54, 22 May 2026 (UTC)

What conventions would those be? Sławomir Biały (talk) 18:55, 22 May 2026 (UTC)

Just for example, which base fields of inner product spaces are allowed? Are regular, normal, etc. spaces assumed Hausdorff? Do complete lattices contain the greatest and smallest elements? Do rings have ones? And so on. 慈居 (talk) 19:04, 22 May 2026 (UTC)

+ Are inner products positive-definite? non-degenerate? Are they left or right linear? 慈居 (talk) 19:07, 22 May 2026 (UTC)

This is a bizarre list of things to worry about. The only item here that seems directly relevant is the linearity convention for complex inner products, and the article is written so that this choice plays essentially no role. It is also explicitly noted in the definition section. Sławomir Biały (talk) 19:15, 22 May 2026 (UTC)

Yes, it is noted in the definition section but how do you know that before reading the definition section? A preliminary section like this (with whatever title) typically writes about various conventions to prepare further reading and it is reasonable to imagine that they are not repeated after this section. 慈居 (talk) 19:35, 22 May 2026 (UTC)

The base field assumption (e.g., whether or not it is complete) is also relevant because it is relevant to the properties of the spaces. 慈居 (talk) 19:39, 22 May 2026 (UTC)

Like I said, the sesquilinearity convention is not important to understand the rest of the article. It is only important in one place, where it is discussed. The article says that the field is the real or complex numbers. I don't see this as an honest point of confusion. Sławomir Biały (talk) 19:56, 22 May 2026 (UTC)

It is not about confusion, it is about inconvenience. The sesquilinearity convention concerns how the concrete inner products are written, it is important for the rest of the article. 慈居 (talk) 20:05, 22 May 2026 (UTC)

I think I am going to disengage with you at this point. It's clear that this discussion is heading nowhere. Sławomir Biały (talk) 20:16, 22 May 2026 (UTC)

Can you explain which part is inconvenient? I don't understand what you're getting at. –jacobolus (t) 23:19, 22 May 2026 (UTC)