Talk:Hilbert space/Archive 2

Error in the introduction
The last edit has introduced an error I think!

The first paragraph reads : In mathematics, a Hilbert space is a to extend concepts from plane geometry to more general settings....

which doesn't make sense. Maybe "is used to extend concepts" ? Can the original editor please clarify... --Nickj69 16:05, 2 March 2007 (UTC)


 * User CSTAR decided to yank the whole passage without waiting for someone to address the issue. The text was the bolded section below, following "In mathematics":


 * a Hilbert space is a to extend concepts from plane geometry to more general settings. Of particular importance is the notion of what it means for things to be perpendicular (or orthognal). The simplest and perhaps the most intuitive example of a Hilbert space is the Euclid Plane. In general...


 * This could probably have been fixed by rephrasing the first sentence as:


 * In mathematics, a Hilbert space is used to extend concepts from plane geometry to more general settings. Of particular importance is the notion of what it means for things to be perpendicular (or orthognal). The simplest and perhaps the most intuitive example of a Hilbert space is the Euclid Plane. In general ...


 * I was glad to see the attempt made to make this article more comprehensible to lay readers, and IMO (speaking as a lay reader) it was a good start. CSTAR, I appreciate your concerns; if you could take a minute to improve on the original attempt, I'm sure we'd all be grateful.
 * * Septegram * Talk * Contributions * 16:42, 2 March 2007 (UTC)
 * That proposed intro paragraph is not a definition, and moreover is not very specific characterization. Lots of things (even lots of mathematical concepts) are used to extend concepts from plane geometry to more general setting. The proper way to fixe this is to say right away that a Hilbert space is an inner product space which satisfies an additional condition.  The proper explanation of the geometrical intuition goes there. --CSTAR 19:08, 2 March 2007 (UTC)
 * Why should the intor be a definition or a characterization. I think the intro should give readers a rough idea of what is going on, then definitions/characterizations come later. Forgive my grammer, I clearly made some mistakes but I was making an honest attempt to make it less technical, the new article isn't an improvement over what was there before.Thenub314 01:47, 3 March 2007 (UTC)
 * I don't know how to say this without sounding rude.
 * "In mathematics, a Hilbert space is an inner product space with an additional property of a technical nature, known as norm completeness. Thus a Hilbert space is real or complex vector space with a positive-definite Hermitian form that is complete in the corresponding norm" would literally be only slightly less comprehensible if it were written in ancient Sumerian.
 * The terms "inner product space," "norm completeness," "real or complex vector space," and "positive-definite Hermitian form" are completely meaningless to someone without (I can only surmise) a serious scientific or mathematical background. To a person who peaked on math in high school algebra, there might as well be a sign saying "Non-experts go away" on this article.
 * I appreciate the attempt, and hope you'll keep slogging at it, but this is still entirely obscure.
 * I tried to develop a suggested replacement in plain English, but simply couldn't figure out where to start.
 * * Septegram * Talk * Contributions * 19:56, 2 March 2007 (UTC)
 * No, it is not saying "non-experts go away" it is saying, "non-experts please go here first."--CSTAR 20:12, 2 March 2007 (UTC)
 * It might as well be.
 * I tried. I went to several of the links from the current introduction.  The results weren't exactly helpful.  Here are some of the introductions, with terms listed in bold that are not particularly informative to the non-expert:


 * Inner Product Space
 * In mathematics, an inner product space is a vector space of arbitrary (possibly infinite) dimension with additional structure, which, among other things, enables generalization of concepts from two or three-dimensional Euclidean geometry. Formally, the additional structure is called an inner product (also called a scalar product). This allows the introduction of geometrical notions such as the angle between vectors or length of vectors. It also allows introduction of the concept of orthogonality between vectors. Inner product spaces generalize Euclidean spaces (with the dot product as the inner product) and are studied in functional analysis.
 * Complete Space
 * In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M.
 * Positive-Definite
 * (Links to "Definite bilinear form," which is equally incomprehensible.)
 * Hermitian Form
 * A Hermitian form (also called a symmetric sesquilinear form), is a sesquilinear form h : V &times; V &rarr; C such that
 * $$h(w,z) = \overline{h(z, w)}$$
 * The standard Hermitian form on Cn is given by
 * $$\langle w,z \rangle = \sum_{i=1}^n \overline{w}_iz_i.$$
 * More generally, the inner product on any Hilbert space is a Hermitian form.


 * (I'm not even going to try to list the incomprehensible parts of this one.)


 * My point here is that much of this material is clearly written for experts, and is therefore useless to a non-expert. I realize it's harder to translate math into English than it is to translate legal into English, but the result is the same: non-specialists haven't the foggiest idea what you're talking about.
 * * Septegram * Talk * Contributions * 20:50, 2 March 2007 (UTC)
 * Well it's certainly helpful to get feedback. Some things clearly can be improved. However, some terms are very basic, such as vector space. I'll think about your points and try to make some changes.--CSTAR 20:55, 2 March 2007 (UTC)
 * Thanks for taking this so well. I know how hard this sort of thing can be; I've worked tech support...
 * When you say "vector space" is "basic," I have to ask how many people on the street you think would understand it if asked?
 * * Septegram * Talk * Contributions * 20:59, 2 March 2007 (UTC)
 * Not many. But probably not many people on the street would ask "what is a Hilbert space"?--CSTAR 21:27, 2 March 2007 (UTC)
 * Touché. Good point; the answer is "Not many, but if they happened to hear about it (say in a science fiction novel, which is where I heard about it) they might want to learn a little more without having to take advanced math courses."
 * * Septegram * Talk * Contributions * 21:38, 2 March 2007 (UTC)

Please have a look at inner product space and continue the discussion over there. Any suggestions are welcome.--CSTAR 21:06, 2 March 2007 (UTC)

I think the first sentance, maybe even the first paragraph should be accessible to middle school students who are given a paper to write about Hilbert's contribution to mathematics. It seems to me this might as well say "If you haven't taken math courses in college, go away". Thenub314 01:52, 3 March 2007 (UTC)


 * Having a definition in the first sentence is a long-established convention here (see Guide to writing better articles), in which we follow other encyclopaedias. The rest of the first paragraph should ideally be widely accessible, though middle school seems to go quite far. It's hard to explain Hilbert spaces if they don't know multidimensional geometry (Euclidean space).
 * On the other hand, I'm not so happy with CSTAR's suggestion that lay readers should just follow the link to inner product space. I tried to rewrite the first paragraph to make it a bit easier for lay readers to understand. -- Jitse Niesen (talk) 11:03, 3 March 2007 (UTC)
 * Your correct. I was being a bit thick.  Good Job with the intro. Thenub314 14:12, 3 March 2007 (UTC)
 * 'Reply to Jitse. I'm not so happy with my suggestion either, but I think we should be very careful how we try to make this better. I think we all will happily take suggestions, but I din't like the edit I reverted because it wasn't a definition.--CSTAR 20:27, 3 March 2007 (UTC)

I'm not sure how much clearer the intro can get without becoming very, very long. The current intro gives you links to the two immediate prerequisites (inner product space and completeness) as well as a motivating example (Euclidean space). It mentions the geometric implications and that as a result of completeness you get some of the benefits of finite dimensions, even in infinite dimensional spaces. I'm new here, but does anyone else think the template could be removed? J Elliot 18:47, 7 March 2007 (UTC)


 * I'd say "No." As I pointed out above, the antecedents are still essentially meaningless to a layperson.  They, too, lack useful introductions that would be useful to a layperson, so they're not much help in understanding this article.
 * I realize how difficult this must be; some concepts do not translate well into English, but I don't think that's a reason to remove the template. I'd rather it just stayed there until someone who can translate the subject matter takes an interest.  At this point, I'm resisting the temptation to go through the links on this article and slap the same template on a bunch of them.
 * * Septegram * Talk * Contributions * 18:57, 7 March 2007 (UTC)


 * I don't agree. If it's impossible to explain the general meaning to a layperson, then it's not reasonable to put the template there. After all, many universities treat Hilbert spaces only in the second year. In any case, I think the template should be put on the talk page, just as Template:Technical. -- Jitse Niesen (talk) 02:43, 8 March 2007 (UTC)


 * I didn't say it's "impossible," I said it's "difficult." Are you asserting that it is impossible to explain this concept in language accessible to a lay person?
 * * Septegram * Talk * Contributions * 15:27, 8 March 2007 (UTC)


 * Yes, I'm saying that I think it's impossible to explain the definition (specifically, the completeness assumption) to a layperson within the size restrictions of a Wikipedia article. Of course, the current article can undoubtedly be improved, and I'd love to be proved wrong, but that's my position. -- Jitse Niesen (talk) 00:33, 9 March 2007 (UTC)


 * IMHO, one should be careful with those template messages. this is by no means expert territory, as Jitse's comment above pointed out. certain background is probably needed to read the article, but not much and not unreasonably so. mathematics is not like, say, dentistry, where the technical vocabulary is mostly just that and a layperson can sensibly expect some kind of loose translation (e.g. "endodontic treatment" = "root canal" = "they probably drill a hole into your tooth and clean out the nerve canals" and "type # mobility" = "the tooth is loose", etc). Mct mht 13:52, 8 March 2007 (UTC)


 * "This is by no means expert territory?" Do you mean that this article is written in such a way as to be accessible to non-experts?  I took the liberty of looking at your userpage, and I have to wonder if perhaps you are sufficiently expert yourself that you don't remember what it's like to not know these concepts?  I recall a friend of mine from college who took an intro calculus class, taught one semester by a professor whose usual purview was advanced higher math.  He would put an equation on the board, show a couple of steps, and then say "And from here it is intuitively obvious that..." and leap to the end.   For him, it was obvious, but the class was left entirely baffled, which is exactly what happened to me when I tried to read this article.
 * Have you read prior posts about the inaccessibility of this article?
 * * Septegram * Talk * Contributions * 15:27, 8 March 2007 (UTC)


 * The problem is, this is NOT calculus. Calculus needs to be taught to a general audience, but real Hilbert space theory isn't taught to anyone besides graduate students and advanced undergraduates.  If this article communicates that there is a sense of geometry and that complete infinite dimensional spaces take on some characteristics of finite dimensional spaces, I think it gives the average reader a headstart on many math majors studying this material.  I think it's easy to underestimate just how far off the beaten path this material is for the layperson. J Elliot 02:19, 9 March 2007 (UTC)


 * I used calculus as an illustration, not an example. Are you agreeing with Jitse Niesen, above, that this is a subject which cannot possibly be even briefly outlined in such a way as to give the layperson the basic concept of Hilbert space?
 * I'm not at all convinced that this article "communicates that there is a sense of geometry and that complete infinite dimensional spaces take on some characteristics of finite dimensional spaces." In fact, I'm honestly not even sure what that phrase itself means.  It feels closer to comprehensibility, but upon trying to rephrase it I find I really don't know what that means.
 * You said "I think it's easy to underestimate just how far off the beaten path this material is for the layperson." For my money, that's all the more reason to put in some language specifically aimed at the unfortunate layperson like me, who stumbles across a reference to the term and comes looking for enlightenment.  There's an old saying that you don't really understand a subject until you can explain it to your grandmother.  Granny's still looking at the intro and scratching her sainted white head...
 * * Septegram * Talk * Contributions * 15:19, 9 March 2007 (UTC)
 * You mentioned that you stumbled across this term in a SciFi book. I'm curious as to why the author didn't say something about the meaning of a Hilbert space there? Do you really need to know any more than that it's a space of some kind with angles and that is used in formulations of Quantum mechanics?--CSTAR 16:10, 9 March 2007 (UTC)
 * I didn't even know that until you said so: as I keep pointing out, the material on this page is unbeleivably obscure. I believe the author mentioned that Hilbert space is a kind of space that has as many dimensions as you need to solve your problem.  The name popped into my head a few days back and I thought "Hey, maybe Wikipedia could give me a clearer idea of what this really is."  Which brought us here...
 * Would it be acceptable to the matherati to have an intro sentence that says something like "Hilbert space is a mathematical construct used in the formulation of some of the more esoteric principles of quantum mechanics?" That might be insufficiently detailed to really describe Hilbert space, but would it be wrong?
 * * Septegram * Talk * Contributions * 17:19, 9 March 2007 (UTC)

Reply The first sentence should remain unchanged, in my opinion. Possibly some such in the second sentence. Note also that "esoteric principles" is definitely not correct.--CSTAR 18:36, 9 March 2007 (UTC)


 * OK, so change "esoteric principles" to something more accurate: I'm just trying to suggest a way to make this article minimally accessible. Unfortunately, the first sentence is still gibberish to someone not versed in the necessary math.
 * I really fear I'm sounding crankier and more obnoxious than I mean to; I hope this discussion doesn't garner me any new enemies...
 * Let me point out that WP:LEAD says
 * [[Wikipedia:Lead_section#Provide_an_accessible_overview|The lead section should briefly summarize the most important points covered in an article in such a way that it can stand on its own as a concise version of the article (e.g. when a related article gives a brief overview of the topic in question). It is even more important here than for the rest of the article that the text be accessible, and consideration should be given to creating interest in reading the whole article (see news style and summary style). The first sentence in the lead section should be a concise definition of the topic unless that definition is implied by the title (such as 'History of …' and similar titles).

In general, specialized terminology should be avoided in an introduction. Where uncommon terms are essential to describing the subject, they should be placed in context, briefly defined, and linked. The subject should be placed in a context with which many readers could be expected to be familiar]]
 * I'm sure you can see how the intro as it stands doesn't really fit that.
 * * Septegram * Talk * Contributions * 19:38, 9 March 2007 (UTC)


 * Well, what are the prerequisites of understanding Hilbert spaces? A reader would at least have to know about (abstract) vector spaces and their subspaces, about inner products, about normed vector spaces (or topological vector spaces), and about Cauchy sequences and convergence. I don't see how all this can be explained within the space of the intro of an article, but I will try: We could describe vector spaces as mathematical objects with just enough structure to allow to talk about the superposition of their elements. Now we can talk about bases as minimal selections of vector space elements that allow to represent all other vector space elements as (unique, finite) superpositions. Now we have to talk about base changes. Now we have to explain how the generalization of finite superpositions of vector space elements (linear combinations) to infinite superpositions requires additional structure, namely some notion of distance/closeness, so that we can define Cauchy sequences, i.e. sequences where the elements get closer and closer to each other the farer we travel into the sequence. Now we have to explain how inner product spaces allow us to talk about projections of vector space elements on subspaces and about orthogonality of vector space elements, and how the notion of an inner product also allows for a natural definition the length of a vector space element, and how the length of the difference of two vector space elements provides us with meaningful notion of the distance of two vector space elements that is also invariant under base changes. Now we have to define closed subspaces, and explain that Hilbert spaces are special in that they always have an orthonormal basis.
 * Would something like this be acceptable to experts and, more importantly, enlightening to laymen? Hilbert spaces really are sophisticated mathematical constructs. The simple examples of Hilbert spaces are "boring" (all finite dimensional inner product spaces are Hilbert spaces), and even the simplest examples of Hilbert spaces that are worth the bother are already highly abstract. &mdash; Tobias Bergemann 21:06, 9 March 2007 (UTC)
 * Wait, wait. Here we go again...
 * I'm sorry, Tobias, but you're using terminology that's very esoteric to the average layperson, probably because you're trying to grant too complete an understanding in the lead section. I'm still thinking of something like "Hilbert space is a mathematical construct used in..." (fill in the blank in layman's terms).
 * * Septegram * Talk * Contributions * 21:51, 9 March 2007 (UTC)
 * Well, to rip off the section titled "Introduction": "Hilbert spaces are mathematical constructs used e.g. in the study of partial differential equations and the spectral analysis of functions, and to model the possible states of quantum mechanical systems." I really do not see how this would help a layman, and these are already the most concrete applications of Hilbert spaces that I can think of, and the most vague descriptions that I would be willing to accept. (I really don't try to be difficult. In fact, after reading the discussion above I have spend a fews hours trying to find words that would explain Hilbert spaces in terms that my parents would understand and am myself frustrated by my failure. It's just that Hilbert spaces occupy a place in mathematics where even the originally abstract foundations have again been abstracted.) &mdash; Tobias Bergemann 22:39, 9 March 2007 (UTC)
 * Well done!
 * If you'll accept
 * "Hilbert spaces are mathematical constructs, the uses of which include the study of partial differential equations, the spectral analysis of functions, and the modeling of possible states of quantum mechanical systems."
 * then so will I.
 * Even if it still needs tweaking, we may be getting close to a solution.
 * * Septegram * Talk * Contributions * 22:54, 9 March 2007 (UTC)
 * The quotation of the wikipedia guideline above ends with "The subject should be placed in a context with which many readers could be expected to be familiar." I just cannot think of any such context wrt. Hilbert spaces. I would expect every first-year student of mathematics or physics or engineering to know enough about vector spaces and Cauchy sequences to be able to understand at least the definition of a Hilbert space. In that sense, and in that sense only, do Hilbert spaces not require expert knowledge. &mdash; Tobias Bergemann 22:50, 9 March 2007 (UTC)
 * Septegram: I think that sentence is a good summary of some of the applications of Hilbert spaces and something like it should probably be in the article, but I'm not sure what it does to address the issue at hand. Is it any better to say that Hilbert spaces have applications to spectral analysis and PDEs than it is to say that it is a complete inner product space? 84.2.156.169 23:50, 9 March 2007 (UTC) (<-- that was me: J Elliot 23:53, 9 March 2007 (UTC))
 * I'm confused. There are already two sentences about applications in the lead section: "[Hilbert spaces] provide a context with which to formalize and generalize the concepts of the Fourier series in terms of arbitrary orthogonal polynomials and of the Fourier transform, which are central concepts from functional analysis. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics." So, what's the point of the discussion here? -- Jitse Niesen (talk) 01:12, 10 March 2007 (UTC)
 * Jitse Niesen, the point of the discussion is that "[Hilbert spaces] provide a context with which to formalize and generalize the concepts of the Fourier series in terms of arbitrary orthogonal polynomials and of the Fourier transform, which are central concepts from functional analysis" is essentially meaningless to a non-mathematician. I realize that may be hard to grasp for someone who'd actually gain something from the rest of the article, but it really is the case.
 * I'm trying to get some movement here toward—at the very least— a single introductory sentence that is not wah-wahwah-wah to the average lay person.
 * * Septegram * Talk * Contributions * 21:39, 10 March 2007 (UTC)
 * J Elliot, yes, it is better to address the applications than to refer to "complete inner product space." "Complete inner product space" means nothing to me, and probably to the vast majority of humanity.  However, quantum mechanics, differential equations, and spectral analysis are terms of which a goodly slab of Wikipedia's audience might at least have heard.  That would put Hilbert space at least into some kind of context for them.
 * * Septegram * Talk * Contributions * 21:39, 10 March 2007 (UTC)

(de-indenting) Okay, so you are looking to replace the sentences about applications. That's fine; it just wasn't clear to me. I'm surprised that "spectral analysis" is deemed a less technical term than "Fourier series". I think that J. Elliot meant to refer to the specialization within functional analysis which studies the spectrum of an operator. The link to spectral analysis is misleading because that article talks about the physical spectrum.

I actually quite like to mention "Fourier series" because it gives a concrete application which with many people who are about to study Hilbert spaces are familiar and because it is a simple idea. Perhaps, part of the problem of the current sentence on applications is the convoluted sentence structure. So, let me give it a try:
 * "Hilbert spaces appear in the study of partial differential equations. They are used to formalize and generalize Fourier series, the process in which arbitrary signals are decomposed in simple sinusoidal waves. Another application is in the mathematical formulation of quantum mechanics, where the possible states of a system form a Hilbert space."

The only technical term instead of "Fourier series", which I tried to explain, is "sinusoidal". This is not too important; it can be deleted or perhaps we can call them "sine waves" if you think that's clearer. Another thing I'm not sure about it whether to use "signals" or "functions". A technical point is that "the possible states of a system form a Hilbert space" is a bit misleading in that it ignores the normalization; perhaps we should reformulate ("Hilbert spaces are used to construct the possible states of a system"), perhaps we just don't care.

Incidentally, I fully support attempts to clarify the context in which Hilbert spaces arise, and I believe this can be done in a way that is more-or-less accessible to the general public. My point was that you cannot expect the whole lead section to be accessible to the general public, because it must contain the definition and the definition cannot be made accessible in a couple of paragraphs. I apologize if I didn't make my position clear, especially if that led to some frustration. -- Jitse Niesen (talk) 02:22, 11 March 2007 (UTC)


 * Hi, Jitse Niesen. Sorry it's taken me a little time to get to this: real life calls at times...
 * I'm not really looking to replace the sentences about applications; I'm just looking to start the article with something that will not have laypeople leaving frustrated. I think a reference to Hilbert Space being a mathematical construct and a couple of uses should be plenty, especially if they're uses that make sense (or are at least familiar) to the average reader.  It's too bad the "spectral analysis" isn't the physical spectrum; that would have made it an excellent example.  I think it best we skip that one.
 * May I tweak your proposal for readability-to-the-average-lout-on-the-street (such as I)? Viz,

"Hilbert space is a mathematical construct used in advanced mathematics. Applications of  Hilbert space include the study of partial differential equations and mathematical formulation of quantum mechanics, where the possible states of a system form a Hilbert space."
 * At the risk of being accused of dumbing-down, I have to point out that most laypeople would not understand how a sinusoidal wave becomes compost, which is what most people think of when they hear "decomposition."
 * * Septegram * Talk * Contributions * 20:52, 12 March 2007 (UTC)


 * As you must have noticed, you're not the only one who can't reply immediately!
 * I tried to incorporate your suggestion in the article. Let me know what you think. -- Jitse Niesen (talk) 06:09, 18 March 2007 (UTC)


 * I'm afraid it's still mostly gibberish to a non-expert. If we can just get the first sentence or two to something the average J. on the street can follow, I'm going to stop being annoying about this.  Unfortunately for your revision, "inner product space that is also a complete normed vector space (a Banach space) under the norm induced by the inner product" contains at least five terms that most people would find utterly meaningless:
 * inner product space
 * complete normed vector space (this could qualify as up to three meaningless terms, depending on how one groups the terms)
 * Banach space
 * norm (in this context, as opposed to others)
 * inner product
 * Do you see why I have a problem with it as it stands, and why I made the suggestion I did above? Would you be comfortable with just sticking "Hilbert space is a mathematical construct used in the study of..." at the very beginning, and then whatever you want after that?  I'd settle for that; I'm feeling a bit beaten down with this wrangling {wry grin}
 * * Septegram * Talk * Contributions * 17:57, 19 March 2007 (UTC)


 * OK, more than a week with no further discussion: I think y'all may be as worn out with this as I am. So I've Been Bold and rewritten the introduction.  Absent any enormous howls of outrage, I'm going to unwatch this page pretty soon and consider this discussion wrapped up.
 * Given my experiences here, I've given up on any plans to make similar overtures at any other mathematically-related articles, desperately though they need it.
 * Someone already removed the "Expert Attention Required" template, so that's a non-issue.
 * * Septegram * Talk * Contributions * 19:41, 27 March 2007 (UTC)

'''OK, I quit.

'''Since CSTAR saw fit to revert my change instead of attempting to improve it, I give up. If someone else wants to try to have a simple introduction intelligible to a lay person, then they are welcome to take over the task of persuading experts to allow one.

'''Alternatively, the experts may sit back and relax, smug in their obscurity.

'''I'm done with slamming my head against this particular wall.

 * Septegram * Talk * Contributions * 21:28, 30 March 2007 (UTC)

Reply: I don't know what you (that is user:Septegram) are talking about. As is easily verifiable by inspection of the contributions list, prior to this edit, I haven't made a single edit to Wikipedia since March 23. The edits you (that is Septegram) seem to find objectionable to were made later by User:Arcfrk. Though I happen to think User:Arcfrk's modifications to this article are pretty damned good, I should clarify that
 * User:Arcfrk $$\neq$$ User:CSTAR

--CSTAR 02:28, 4 April 2007 (UTC)


 * I apologize. I misread the notation here.  The reversion was done by Linas to a version of yours.
 * The rest of my comments stand.
 * Apparently certain sections of the MOS are superseded by the addition of a nifty template. I thought that the    template more appropriate, but when I added it, it was promptly removed.  As I said, I give up.
 * * Septegram * Talk * Contributions * 14:17, 4 April 2007 (UTC) (disgusted)

Some thought on improving the introduction
It is obvious that many well-meaning people are engaged in vigorous debate concerning improving the introduction to this important article. It is equally obvious that they are not succeeding. I read through most of the discussion (although it tends to be quite repetitive...) and would like to offer several observations.
 * There is a clear disagreement on the appropriate level of the introduction. Some people just want to know "what is it all about" and others yearn for technical terms thrown at them right off the start. It will be essentially impossible to reconcile the two points of view, especially if all sides hold their ground.
 * Sometimes rewriting an article (a paragraph, an introduction) completely will produce a much better result than doing many peacemeal iterative edits, just pushing the sentences around without resolving the fundamental contradictions. What may be needed is a fresh look.
 * Ask yourself: "Am I competent enough to write about it? Do I understand this really, really well? Do I care more about getting wonderful result in the end or having participated in the process?"

Here are some possible ways to address those points, call them suggestions if you wish.
 * It is not necessary to put all you know about Hilbert spaces in the introduction. An interested reader who is not afraid of math will read on anyway, so much of this material can be moved further into the body of the article. For example, a section titled "Motivation" can be added right after the introduction, and much of the technical sounding material can be moved there.
 * If the introduction is sufficiently broad and non-technical, many more people will continue to read the article. As a complement to the preceding remark, kind of a converse statement, a person generally unfamiliar with the material may become intimidated and never go beyond the first paragraph unless the article starts very, very gently. There are very high-level reasons why Hilbert spaces are interesting that do not require explaining first what they are, hence would not produce the intimidation effect. One can talk about their history, or in very general terms, how they brought together several areas of mathematics, or what other sciences use them.
 * Introduction doesn't have to be long. It should highlight the topic, whet the reader's appetite, not glut him with heavy food for thought, so to speak.
 * Here is another example of a high-level issue that can be briefly addressed in the beginning: To what extent Hilbert space theory influenced or still influences the development of mathematical thought?
 * If the technical terms are to be used at all, it's good to use intuitive sounding terms like angle and distance, and probably OK to mention that most Hilbert spaces are infinite dimensional and composed of functions, but quite useless to talk about orthogonal projections onto a closed subspace.
 * On the other hand, encyclopedia is not a substitute for proper course of study! Most abstract mathematical concepts are, well, hard and abstract. If it took scientists thousands of years to arrive at them, maybe it's unreasonable to expect that you will "get" them by reading an encyclopedic article. Think about it.
 * It may be helpful to take, let's say, a week's break from the editing and let the dust settle. During this vacation you can take a look at how other Wikipedias (Simple English, Spanish, German, Russian, ...., make your pick) deal with those pesky issues. How do Encyclopedia Britannica, American Heritage Dictionary, various online mathematical encyclopedias approach it?  How would Albert Einstein explain what a Hilbert space is? John Von Neumann? Paul Halmos? It may be a very revealing experience.
 * Last, but not least, enjoy your time at Wikipedia! Make it a pleasant and useful exprerience for yourself, as well as for the others. Arcfrk 10:49, 10 March 2007 (UTC)


 * Your comments apply to many of our articles, especially ones of interest to a wider audience. We do have writing tips for mathematics, which say much the same thing. Across Wikipedia, visibility is correlated with more edits; within an article, the first paragraph and first sentence often get far more attention than the body. A typical tension is between the "lead with a formal definition" school and the "begin with intuition" school; our official recommendation is the latter.
 * I approve of the competence question (mostly), but many editors seem to think that's un-Wikipedian. After all, the front page says boldly, "Welcome to Wikipedia, the free encyclopedia that anyone can edit." There is no demand for subject knowledge, nor English-language fluency, nor writing skills, nor social graces. Sometimes it shows. --KSmrqT 17:11, 16 March 2007 (UTC)

Beginning of the clean-up
Well, I was really hoping for a while that this article gets to a decent state without my participation, but it was not meant to be (I am not implying, by the way, that it would get into that state even with my contribution ). After a recent major war of words and minor revert hostilities, I've taken the liberty of pushing the technical aspects down below the lead, and adding a section on intuitive meaning. Some of the material is in the process of rearrangement, naturally, it introduces temporary redundancies. Stay tuned! Arcfrk 00:30, 31 March 2007 (UTC)

Cleaned up a bit more, now the section on applications appears to be thin and out of place. I don't have time to finish today, so feel free to edit or copy edit! Arcfrk 05:17, 31 March 2007 (UTC)


 * I have reworded the intro paragraph. Many of us use popups so we can see the first paragraph of an article by hovering over a link. This is a considerable time-saver! Therefore it is preferable to retain a concise formal statement in the paragraph we will see as a popup, though there is no need for it to be the first sentence. --KSmrqT 01:35, 1 April 2007 (UTC)

I'd like to reiterate my earlier request to other editors: please, keep the lead relatively low-level, so that an interested person can continue reading without being knocked out unconcious by the first paragraph! After all, Hilbert spaces are not motives or Ito integral. If we claim that they are widely used in physics and engineering (by the way, it would be nice to give an example of that - would signal processing do?), we ought to be able to explain them to people in those fields. Arcfrk 12:55, 1 April 2007 (UTC)


 * Starting at the end, with an example: Signal processing and control theory come in sampled and continuous versions, with connections between them. The concept of a Hilbert space allows a unified approach. For example, if we have a signal in the form of 8 samples,
 * $$ (x_0,x_1,x_2,x_3,x_4,x_5,x_6,x_7), \,\!$$
 * we can construct an orthonormal basis for this 8-dimensional real inner product space using sinusoids of differing frequencies. More conveniently, we treat the space as complex, with the standard complex inner product, and use a complex exponential in place of a sine or cosine.
 * $$\mathbf{b}_k = (e^{0 i \frac{2\pi k}{8}},e^{1 i \frac{2\pi k}{8}},e^{2 i \frac{2\pi k}{8}},e^{3 i \frac{2\pi k}{8}},e^{4 i \frac{2\pi k}{8}},e^{5 i \frac{2\pi k}{8}},e^{6 i \frac{2\pi k}{8}},e^{7 i \frac{2\pi k}{8}}) \,\!$$
 * Then the discrete Fourier transform is merely a change of basis from the impulse basis to the frequency basis. The continuous Fourier transform is the same thing in an infinite-dimensional Hilbert space. Very convenient, much used. Is this the kind of example you want?
 * I'm confused by your first admonition. Are you saying that my rewrite was too mathematically demanding? Please clarify.
 * One emphasis that has gone missing in your revision of my rewrite is that this is all about infinite-dimensional spaces. I'll guarantee you the average reader will accept the idea of counting one, two, three, …, infinity more easily than some bizarre abstract notion of "function space"!
 * Also missing in the revision is my explicit statement motivating completeness, the pivotal new feature which is otherwise just another mathematical mystery.
 * I'm happy to see that some of what I wrote met with approval (apparently), and I'm curious to hear comments about the points I raise.
 * More broadly, is my rewrite, and Arcfrk's revision, a move in the right direction, or simply more word hash? --KSmrqT 19:04, 1 April 2007 (UTC)


 * Yes, I did think that your lead was emphasizing technical details a bit too much, at the expense of giving a broader picture of the subject. Nevertheless, I tried to keep changes to the minimum. There is already a sentence in the lead explaining the intuitive meaning of completeness, and any extended discussion can (and probably should) be postponed until the formal definition is given, so I moved it there.


 * To address your comment about infinite-dimensionality: you are too charitable in assuming that people are comfortable with idea of infinity. Both history of mathematics and popular literature indicate that quite the opposite is true. But in this case, we are talking about something qualitatively more involved: infinite-dimensionality. There is the usual quote about how dimension is one of the deepest properties of space; for most people, even within mathematics, the meaning of dimension is far from intuitively clear. Indeed, within linear algebra it only comes after a fair amount of preliminary work. In any case, my feeling is that the phrase "infinite-dimensional space" is utterly meaningless without context, and since for people not already conversant in linear algebra the context is absent, it is utterly meaningless for them. To provide the proper context even in a full length encyclopedic article is challenging at best; in an introductory paragraph it's all but impossible.


 * A statement such as
 * Hilbert space theory is all about infinite-dimensional examples
 * is unsatisfying in at least two respects: one, it implicitly assumes that the reader already has some idea of "finite-dimensional examples", which is rather unreasonable if the reader is just trying to learn what this is all about; two, it definitely skews the perspective, because even though Hilbert spaces used in partial differential equations are, for the most part, infinite-dimensional, nevertheless, Hilbert spaces (or inner product spaces, if you want to insist on the distinction) have useful applications that do not require dimensionality considerations whatsoever and many techniques have direct geometric meaning (e.g. direct sum decompositions and projections). I would agree that the definition of Hilbert space is "about infinite-dimensional spaces" in the sense that completeness is automatic for finite-dimensional spaces, but this is a rather subtle point and does not directly relate to applications.


 * I do not think function spaces are intuitive or good to give as examples of Hilbert spaces. Functions, on the other hand, are more concrete objects than even vectors for most people using mathematics, they bring out the point well, and are therefore good to mention. Arcfrk 22:58, 3 April 2007 (UTC)


 * Let's start with the issue of function spaces. I inherited this example from you, and you kept it from your predecessors. Do you really think "space of functions" will make sense to a reader if "function space" does not? I find that hard to accept. There is a huge difference between a function, say f(x) = 3x2, and a space of functions. (Note that you link to the former, which is irrelevant and useless, whereas I link to the latter, which may be of help.) Also, I kept the example only in a role where it would do little harm to overall understanding if it was a mystery, whereas you have introduced it early and in an essential role. I think that is a mistake.
 * The idea that space is three-dimensional and a plane is two-dimensional is not some unfathomable deep property of mathematics, beyond the imagination of the average reader; nor is the idea of counting as high as we like. This is all that I ask of the reader, a modest stretch of the imagination. For these readers, your rewording has made the article less accessible, not more. In fact, I went back and read the intro before you touched it, and was surprised to find that it was better still — except for an opening sentence or two that was a technical definition.
 * We seem to have rather different intuitions about what is too technical and what is more helpful for a lay reader. Worse, I had assumed that your original edits improved the introductory paragraphs, and now I find that is not so. Thus I conclude that we are just thrashing rather than improving. Therefore, I am going to withdraw my hand with only a few fingertips missing. (Declares victory and strategically redeploys.) --KSmrqT 00:38, 4 April 2007 (UTC)

Template
Apparently I'm not the only one who thinks this article is essentially incomprehensible. I added the template in the hopes of getting some expert(s) to work on it so that the average reader can get even a small grasp of what it's about. Following the links in the template will give some information on what might be done to improve the article's accessibility. As the link points out, this is not a request to "dumb down" the article, but rather to add some text that is meaningful to someone without the necessary mathematics to understand it as written. I suppose one might gain some understanding of the subject by following every link (and the links therein, ad infinitum), but one might not, too. I counted easily a dozen terms in the introduction that would require research on my part to grasp, and while my math studies pretty much stopped in high school, I am not exactly uneducated. When the intro is beyond a pretty average person's comprehension, something is at least a bit broken.
 * Septegram * Talk * Contributions * 14:59, 1 March 2007 (UTC)


 * The core problem, all too common even in many college science texts, is an inability to write or translate concepts clearly, or at least in a reasonably comprehensible manner. Of course, given the level of abstraction of the subject-matter and the limits of linguistics, this is not easy.  But it is necessary anyway.  It is, BTW, the "secret" of the Dummies series of books. Tmangray 20:31, 11 April 2007 (UTC)


 * Concur. I'm pretty good at doing that in fields where I have some expertise or experience, but (obviously) this one is wa-ay outside my realm of understanding.  That's why I asked for one pitiful sentence to start the article that would be meaningful to a layperson.  Apparently, that's a hopeless cause.
 * I've pretty much given up on this article. I keep watching it out of morbid curiosity, I suppose.
 * * Septegram * Talk * Contributions * 21:05, 11 April 2007 (UTC)


 * Septegram, I think you are a bit unfair to many people who worked to make this article, and the introduction in particular, accessible. Right now there is a very clear disclaimer at the top about "prerequisites" for complete understanding. Contrary to your impression, the first sentence of the lead does a good job of conveying the meaning of Hilbert spaces without technical jargon. It is ok to be unknowledgeable about a topic; to flaunt one's ignorance is a different matter altogether. Incidentally, some highly qualified editors have stopped contributing to fundamental articles, like this one, because they feel that between reverting crank contributions and being accused of delibarately imposing suffering upon the readers, they can find better uses for their time. Which is a pity! Arcfrk 22:40, 11 April 2007 (UTC)


 * I appreciate your position, Arcfrk, but the problem is that the introduction is not accessible. I saw the disclaimer, but that doesn't make the introduction (let alone the article) accessible; instead it essentially says "it is not possible to make this article accessible."  Frankly, I find it hard to believe that it is impossible to make at least the introduction accessible to someone who is not familiar with advanced mathematics.
 * And, contrary to your impression, the lead does not convey the meaning of Hilbert spaces to the average reader off the street.
 * I resent the accusation that I am "flaunting my ignorance." I am a reasonably well-read and -educated individual, probably better so than the average joe on the street; certanly not less so.  The introduction does not explain to me what Hilbert spaces are about; I think it likely that it would not do so for most people.
 * If you have read the discussion on this subject, I'm sure you will note that I did not wander in here and say "You people suck, and you're trying to hide the meaning of this article from me." Instead, I invited assistance from experts in making the article more comprehensible.  Despite the best efforts of several people, the article has not become noticeably more comprehensible than when I started this discussion.  I appreciate that, but I also feel that there's an undertone of snobbery here, and it is to that which I object strenuously.
 * I have cited Wikipedia's policies, to no avail.
 * I agree that it is unfortunate that qualified editors have left, but if they are not able to fulfil Wikipedia's need for accessible articles, perhaps they are/were not the best ones for the task. There's a saying that you don't really understand something until you can explain it to your grandmother.  I realize that's an oversimplification, but I also feel that my attempts to improve the article were resisted, and the final straw was when my suggestion for an introductory sentence was peremptorily reverted, rather than improved.  Hence my "I give up."
 * * Septegram * Talk * Contributions * 14:31, 12 April 2007 (UTC)


 * A key part of the problem lies with the understandable bias for precision on the part of those already well-acquainted with the subject-matter. It's sort of the science equivalent of political correctness.  Yes, precision is good---ultimately--- especially in science, but one has to build up to it in order to effectively communicate.  And one has to be good at communication per se too.  It's obvious to those who already know something about these topics that the ideas are not being stated entirely properly even in those passages which are attempting to convey a complex idea precisely.  The recourse to  impressive mathematical formulas is not a substitute for text.  Experts attempting to communicate complex ideas need to curb their bias to absolute precision, and their impatience, and hewn their writing skills, if they wish to be effective. This can be as much art as science. Tmangray 05:36, 12 April 2007 (UTC)


 * I've heard enough of this nonsense. If the only mathematics one has mastered is pre-university level, perhaps the quadratic formula, then it is absurd to extrapolate that experience to all of mathematics. It is simply false that "if only someone would explain it well it would all be understandable to anyone". Some topics are difficult. One source of difficulty in mathematics is layering, so that the mere definition of an advanced topic requires first mastering difficult prior topics. Hilbert spaces were created as a way to formalize a geometry with infinite dimensions. Not two dimensions (spherical geometry), not three dimensions (Euclidean space), but infinite dimensions. To explain why this is of interest, to explain what dimension means in a way that extends to infinite dimensions, and to state a definition of a Hilbert space, we must draw on more than a lay reader will know. Sorry. Too bad. Get over it. If you want to learn the topic, do the work. Thousands of years ago a mathematician said the same thing to a great ruler: "There is no royal road to geometry." And Einstein, who did his best to explain his theories to the general public, said "Everything should be made as simple as possible, but not simpler." What you see here is the simple explanation.
 * Perhaps as consolation we could suggest a correspondence course in neurosurgery. --KSmrqT 06:33, 12 April 2007 (UTC)


 * To what "nonsense" do you refer, KSmrq? To the notion that the lead sentence in an article might give we poor benighted non-mathematicians a basic notion of what is meant by "Hilbert space?"
 * "Sorry. Too bad.  Get over it." is not an acceptable approach to creating an encyclopedia.  What I see is not a sufficiently simple explanation; even Einstein's work has been expressed in ways that laypeople can get at least a basic grasp of them, which is more than can be said for this article.  Your attitude is exactly the one to which I have referred before; the idea that even the most basic concept of an article is beyond those who have not studied higher mathematics.  This is not an encyclopedic approach, but one more suited for a textbook.
 * However, you may have actually solved the problem, if this statement is accurate:

Hilbert spaces were created as a way to formalize a geometry with infinite dimensions.
 * This would make a pretty good introductory sentence, in fact. I'd modify it slightly, poor benighted layperson that I am, to something like

In mathematics, Hilbert spaces are a way to formalize (maybe I'd use "describe") geometry in a context of infinite dimensions, rather than the two or three with which we are usually familiar.
 * Is there any reason why this is not a suitable intro sentence?
 * Anyone?
 * * Septegram * Talk * Contributions * 14:38, 12 April 2007 (UTC)


 * I vote for it. I vote for more like it.
 * The point is not to simplify the whole article, but to build up to the complexity. Certainly, links are one way to do this, but there should also be comprehesible introductory material summarizing what these links say within the body of the article itself. Tmangray 19:56, 12 April 2007 (UTC)

Unindent I agree, Tmangray. I certainly don't want the whole article simplified/dumbed down to the point where it's not useful to experts in the field or people who are studying the subject. I hope I didn't give that impression.

Does anyone have a better word than "formalize?" Does "describe" work as well, for the sake of laypeople?


 * Septegram * Talk * Contributions * 20:14, 12 April 2007 (UTC)


 * As I've said before, you have to be way more specific about what the problem with the current version is. You just say "inaccessible" without telling us what you don't understand. The problem with your proposal is that it's extremely vague. There are a number of settings in which elementary geometry can be extended to infinite dimensions: vector space, normed space, Banach space and inner product space to name a few relatively easy ones. The WP:LEAD guideline says that the first sentence should be a definition and your proposal is so imprecise that it's not even close. It's ironic that you accuse others of ignoring Wikipedia's policies which you cite, while the only thing you cited is WP:LEAD, not a policy but a guideline, and you're not obeying even the part that you cited. Now, I don't think that's necessarily a bad thing - in fact, I think it's impossible to adhere to all aspects of that guideline in this article - I just want to warn you tat it would be more productive if you were more careful when making accusations.
 * In advice you praised, we were told to take a look at how other encyclopaedias approach the subject. Well, have a look at the article in EB and say how you like it. I think it takes a strange approach (it describes what is actually just one example of a Hilbert space, and says only at the end that the definition is broader), but perhaps that helps. Can you find any other general-purpose encyclopaedia with articles on Hilbert space? Encarta doesn't seem to have one. -- Jitse Niesen (talk) 01:05, 13 April 2007 (UTC)


 * I should add that I like your current suggestion much better than the previous one, where you proposed focussing on the applications in the very first sentence. -- Jitse Niesen (talk) 13:31, 13 April 2007 (UTC)


 * *sigh*
 * I've explained in the past what I meant by "inaccessible." Look at my response above that starts with "It might as well be."  If you think "vector space, normed space, Banach space and inner product space to name a few relatively easy ones" are "relatively easy ones," then it's pretty clear that you have no comprehension of how a layperson would feel upon looking at that sentence.  Not one of those terms is likely to be in a layperson's lexicon.
 * My proposal grows out of a remark above by KSmrq. I reworked it slightly for the benefit of we poor benighted laypeople, but it would probably work pretty well as it stood.  I'm not making any pretense of it being complete or comprehensive, just comprehensible and, if the Gods are kind, not actually incorrect.  If it's comprehensible and not incorrect, I think it'll do.  Your mileage may vary.
 * You're right: WP:LEAD is not a policy but a guideline. I misspoke (miswrote, if you want to be precise).
 * If you don't think I'm following the part I cited by attempting to make the f*****' introductory sentence "written in a clear and accessible style," then I really do give up.
 * I looked at the Britannica article. The intro sentence, "in mathematics, an example of an infinite-dimensional space that had a major impact in analysis and topology" is at least meaningful to a layperson.  Some of the versions we've had up there have been utterly useless for that purpose.  You may think it's a strange approach, but at least it leaves the lay reader with some idea (however pathetically feeble and incomplete) of the subject under discussion.
 * Unwatch.
 * * Septegram * Talk * Contributions * 14:18, 13 April 2007 (UTC)
 * Because I really do care about Wikipedia, I want to stop by and mention that a friend of mine suggested a separate introductory article be created, similar to the Introduction to quantum mechanics. I'm not sure if this is practical, but it would be a viable solution iff someone could be found who could write it and cared enough to do so.
 * * Septegram * Talk * Contributions * 15:34, 16 April 2007 (UTC)

Good Work So Far
Not all writers are knowledgeable in advanced mathematics. And not all mathematicians are good at writing. I think this is a pretty good start, people. The important thing is to not give up and keep working at it -- in a spirit of cooperation. Good luck. —The preceding unsigned comment was added by Edwardpiercy (talk • contribs).

GA on hold
This article has been reviewed as part of WikiProject Good articles/Project quality task force. In reviewing the article against the Good article criteria, I have found there are some issues that need to be addressed. References are not specific and not enough. I am giving seven days for improvements to be made. If issues are addressed, the article will remain listed as a Good article. Otherwise, it will be delisted. If improved after it has been delisted, it may be nominated at WP:GAC. Feel free to drop a message on my talk page if you have any questions. Regards, OhanaUnitedTalk page 21:40, 8 September 2007 (UTC)

I have no idea what this "task force" is, and don't recall any indications of any type of ongoing reviewing process for this article. But since when have issuing ultimatums become an acceptable norm of behaviour on Wikipedia (even if it is followed by hollow "Regards")? Why do you feel that anyone editing this article owes you anything and why adopt such a patronizing attitude? Arcfrk 22:17, 8 September 2007 (UTC)


 * In reviewing the article I have found that the references are quite satisfactory. I therefore as of now issue an indefinite reprieve for the article of the above ultimatum. --Lambiam 04:17, 9 September 2007 (UTC)
 * Well, one good thing we can say about this task force is that it doesn't have any armaments.--CSTAR 06:02, 9 September 2007 (UTC)


 * Arcfrk, this is a new reviewing process (called the GA Sweeps) because many editors felt that GA process is not good enough. To ensure the article remains in good condition after it passes the GA process, we conduct reviews to all GA because this is the first time. (Yes! We are going to check through every single GA article, that is about 2,800 articles!) We're not "delist-happy" people. Our true intention is to uphold the standards of GA. OhanaUnitedTalk page 14:34, 9 September 2007 (UTC)


 * The mathematical statements in the article, which are absolutely standard theory, can be verified from basically any textbook on the subject, including the two references at the end. The non-mathematical statements have inline "footnotes". What else is required? Are there any contentious claims that are likely to be challenged? --Lambiam 17:33, 9 September 2007 (UTC)


 * The article is good. The Dieudonné reference is good, probably good enough for almost everything in the article. Now, if this is not good enough for GA those concerned are probably looking in the wrong place for relevant issues. Which is common enough here, but I'm against 'initiatives' that can't distinguish real work from make-work. Charles Matthews 19:17, 17 September 2007 (UTC)

In an ideal wikiworld, all articles which meet the good article criteria would be good articles and all article which don't would not be. That is only equitable. The fact that it is easy to list or delist good articles helps make this possible, but still GA is a long way from the ideal. Consequently, some GA reviewers are currently going through all the good articles to check that they make the grade.

This article is a great piece of work, and I am not surprised that editors have been pissed off by tactless ultimatums. The tone of these messages has now been changed so that they are more constructive. There have been many changes at GA. In particular the criteria for inline citation have been relaxed to focus on the main issues of quotations and controversial statements.

Still, I find this article is not as well sourced as it could be, and am not convinced it meets the scientific citation guidelines. How does the reader know that Dieudonne is the place to look for most of the material in the article? The exposition also has a textbook feel in places, especially the motivation section. This makes for a nice read, but isn't particularly encyclopedic, especially if the exposition is unsourced.

I'll try to fix some of these things myself. I have the feeling that Harvard referencing would work quite well here, because many of the current cites actually name the source in the text. If you disagree with my edits, let me know here. Geometry guy 19:47, 20 September 2007 (UTC)


 * This article may be a great piece of work (especially given its size), but I agree with Geometry guy that it's not particularly encyclopaedic. On a cursory look, here are some immediate problems:
 * lack of high level explanations of applications of Hilbert spaces (just listing the subjects where Hilbert space techniques have been applied provides only temporary relief);
 * nonstandard and somewhat inconsistent notation (e.g. Vperp instead of the usual $$V^\perp$$);
 * almost nothing is said on the spectral theory;
 * overall, there is no unifying structure (understable for a wiki-article, but needs to be addressed).
 * On the other hand, I disagree that citations are a weak spot. They are probably two degrees of magnitude less important than other things that need to be done, and at the moment are more than adequate. Are we adopting the model that each article should list books "to look for material in the article" rather than explain that material well itself? Unfortunately, as much as I'd like to see a meaningful GA process, the reviewer's comments above just reinforce the impression that people in that project are simply unqualified to review technical articles, and consequently concentrate on purely mechanical features (such as number of citations) without due consideration of their importance or even meaningfulness. And they try to overcome their lack of expertise by making vague pronouncements concerning perceived problems and by adopting a bossy attitude towards the writers. Instead of creating an impression of authoritativeness, though, it just comes off as incredibly rude and obnoxious. Arcfrk 01:46, 21 September 2007 (UTC)
 * Good points, all. Although it should be noted that other relevant material (spectral theory for instance) is in other articles. A problem that will need to be addressed at some point is how to reasonably (pedagogically, notationally, stylisticlly) integrate all this stuff scattered in various articles.  All this should be focused independently of the Good Article Strike Force or whatever it's called.--CSTAR 04:21, 21 September 2007 (UTC)

Comments
Comment: I claim neither expertise in mathematics nor physics, although have training in both fields, and various unmentioned credentials adequate to offer these comment. Thank the authors for the hard work on this article, which I agree is of significance to the fields of mathematics and very much so to physics. I was informed by the article but also (respectfully) somewhat put off by the article on a variety of points of perspective, scholarship, and obscure communication.

First I agree with the high school mathematician who here complained that the level of abstraction used has reduced the article to nearly incomprehensible for non-experts. I opine that Hilbert's insight is NOT too advanced to explain in plain language. So, my mathematical friends, I ask that you please try harder. Precision need not be sacrificed, but the most efficient way of saying something is not necessarily the most articulate. [If you would like an example of an advanced topic offered in terms even a high school algebra student could understand, I offer up the English translation of a 1905 paper by one A. Einstein concerning moving bodies. There's your example, measure up to that.] Perhaps an example (polynomials?) could be used to illustrate and make concrete the generalizations and abstractions.

With regard to mathematicians inventing quantum mechanics, which is the view communicated by the article, I was reminded of an undergraduate quantum mechanics class of long ago taught by a young, but highly qualified PhD specialized in the field of QM. In addition to physics students the instructor's reputation attracted 4 graduate mathematicians who sought to audit, giving the impression to all that they believed the course somewhat beneath their abilities, which was agreeably conceded with respect to the mathematics, the language spoken in the class. Yet no mathematician finished the course, and I was privately informed they dropped due to inadequate performance on the exams. The mathematics were beneath them, to be sure. Instead it was PHYSICS that formed the framework of the subject which, in arrogance and ignorance, they had omitted from their study. This had doomed them. Beware their lesson.

Mathematics is the language of Physics, and without great mathematicians (such as David Hilbert) our physicists would be mute or reduced to inarticulate grunting. Still, it does no disrespect to David Hilbert (who informally studied physics) to explain that he did not (and never claimed to have) create or invent quantum mechanics, or to remind that quantum mechanics is not creature of mathematics, nor momentum a creature of Hilbert spaces, these are the creatures of PHYSICS, created by Physicists in a line of philosophical thought and experiment from (Aristotle to) Newton to Rutherford to Max Plank to Danish Physicist Neils Bohr (who recieved the Nobel Prize in 1922 for his work in this field). It does no disrespect to David Hilbert to make this clear. It was not made clear, the opposite is arrogantly (but perhaps unintentionally) implied.

In sum, this article is important and should communicate in multiple fields and to more than experts. (That doesn't say it can't also communicate to experts but reminding, experts probably already own the book). The authors are commended for their work, but may not have taken to heart some valid and genuinely offered comment from non-exerts. I have made no changes but encourage review in light of these comments. Thank you. -Pie —Preceding unsigned comment added by 64.142.66.123 (talk • contribs) 22:56, December 23, 2007 (UTC)


 * I took a brief look at the contents of the page, and I see no problem with it overall. The article is a bit rough around the edges, but it makes a decent attempt to explain the intuition behind Hilbert spaces. Anyone reading this page should at least have some background in calculus and linear algebra. Otherwise, why are you here?


 * I reject the notion that we have to simplify the discussion of these technical subjects to the point where anyone with a highschool degree can understand them. Find me a highschool student who actually needs to know about Hilbert spaces.


 * As for your example of the arrogant math grad students auditing the quantum mechanics class: They were probably taking the class because some intuition about quantum mechanics is necessary for understanding a lot of hot topics in mathematics. However, most graduate students are way too busy reading papers and writing their thesis to take a class with weekly homework assignments and midterms. It doesn't sound at all to me like these students were arrogant, or thought that the course was beneath them. It sounds like they were hoping to gain some intuition about their subject, but calculated that full participation in the course was a bad budget of their time. I see people making similar calculations all the time, and the decision to drop the physics activity in favor of the immediate necessities of writing a thesis, etc., tends to be made with a sense of regret. So don't be so hard on the math people :) Miaka314 (talk) 01:16, 6 January 2008 (UTC)

Identified an Obvious Point of Confusion
According to A Dictionary of Physics, Oxford University Press, 3rd Ed. 1996, from the Hilbert space entry "...The dimension of the Hilbert space has nothing to do with the physical dimension of the system..." The wikipedia entry never seems to make this clear. That is, a Hilbert space is an informational linear algebra "space" that has nothing to do with Gallilean-Newtonian 3 dimensional physical space (4 dimensions in Einsteinian spacetime). --Firefly322 (talk) 16:36, 24 February 2008 (UTC)

Suggestion(s)
I am an engineer who took linear algebra and differential equations, and I cannot figure out from this article anything except (maybe) that a Hilbert space is a vector space with infinite dimensions. I have no idea why this would be useful to anyone, because I don't see any examples explained. The so-called examples only seem to be definitions of the inner product with no explanation, and I've never seen complex numbers involved in inner products, so this stuff is pretty incomprehensible.

The 'applications' section is merely a list of fields where Hilbert spaces are used, with no explanation how or why they're used, or how or why they're useful.

Fundamentally, the article is mostly a jumble of facts, with no explanation as to how the facts are relevant or what they mean or how they work. This is the same with almost all other wikipedia math articles, which is why they are all almost completely useless even to me, having a good basis in undergrad math for engineering. Furthermore, as some have pointed out near the top of this discussion, if you jump to any of the terms in the intro to learn what they mean, you end up on a wild goose chase - the articles are all circular and never get to a definition you can sink your teeth into. It's like endlessly following fake porn links.

If you want to make what would probably be the first ever readable math article on wikipedia for anything beyond high-school math, I would suggest _at least_ trying to explain it in terms someone could understand coming from non-complex linear algebra.

It might help to think about when and in what context Hilbert spaces are taught. I have no idea what sort of course would teach this - maybe some random graduate mathematics course, or maybe people get a crash course when they take some graduate physics classes? What are the usual prerequisites to learning this subject matter? If you know the prerequisites, you can attempt to explain it in terms of those prerequisites as a first try. Torokun (talk) 19:09, 28 February 2008 (UTC)


 * Yes, this article is very incomplete as far as applications go. It presents mathematical theory of Hilbert spaces, it talks a bit about why they are used, but doesn't really show how. You may find some applications at Fourier analysis and Mathematical formulation of quantum mechanics.


 * You can easily fill up a book by a single example explained from "first principles". Hence it's inevitable that links will be used; since wikipedia is a continuously evolving project, not all of them have been perfected and coordinated yet (and may never be &hellip;), that requires an enormous amount of work under any circumstances, and it is most challenging in a multi-author, volunteer-based environment (by the way, making deprecating comments about their output is counterproductive, as it is likely to turn people away from improving the articles). However, I am a bit puzzled by your two comments:


 * Definition. There is a formal definition of a Hilbert space in section "Definition". It is preceded by a non-technical definition in the lead and the comments on the intuition behind this definition.
 * Prerequisites. You have complained that prerequisites are not clear. But in the second line, you can clearly see the prerequisites, and yes, they involve linear algebra.
 * Arcfrk (talk) 22:45, 28 February 2008 (UTC)


 * From time to time I read the complaints posted here about this article. Having been on the receiving end of some of these complaints, I am reluctant to say anything even mildly positive about the article. Let me first agree that there is always room for improvement and usually something constructive can be gleaned from any post. Torokun's points seem fairly mild as complaints go. But I am particularly puzzled by his/her last remark, that it is unclear what kinds of courses one would cover this material. Is that really true? Wouldn't it be clear that the material is needed for instance for any reasonably systematic presentation of quantum mechanics or in a moderately advanced course in partial differential equations or in any introductory course in functional analysis?


 * Maybe it might be helpful to include a list of specific university courses where this material is covered.--CSTAR (talk) 08:03, 29 February 2008 (UTC)

Lebesgue spaces
The only problems that I had with a recent edit is that A) it switched notation style mid-section, B) bundling the qualifier in with the "resulting space is complete" comment added a note of ambiguity that I didn't feel was made clear by the bracketing commas. I don't have a problem with mentioning the weakness of Riemann integration here, and apologise for the mess caused by edit conflicts. --Sturm 13:13, 29 February 2008 (UTC)


 * No need to apologize. You're right about the notation.  I hadn't even realized it!  Cheers, Silly rabbit (talk) 13:27, 29 February 2008 (UTC)

Unitary space
The section on Definitions and examples states that, "Older books and papers sometimes call a Hilbert space a unitary space or a linear space with an inner product, but this terminology has fallen out of use." Yet here are details of a book, published in 2006, which uses the term "unitary space" to describe complex Euclidean space. So that quoted sentence needs correcting. Also, I don't think that the Unitary space page should redirect here - it should be an article in its own right. -- Cheers, Steelpillow 06:14, 27 May 2008 (UTC)


 * I believe that "Has fallen out of use" is an accurate assessment. It does not mean that no one uses it, rather, that the terminology has gradually become obsolete. Your example is unconvincing, firstly, because it's a single book, and more significantly, because it's a reference (i.e. compilative) text written by non-mathematicians, who are moreover not native English speakers. To the best of my knowledge, it is not a "standard" reference, like Abramowitz and Stegun or Gradshtein and Ryzhik (in a different field), so it's unlikely that its choice of terminology is indicative of a widespread pattern. Arcfrk (talk) 21:09, 29 May 2008 (UTC)


 * Thanks for the enlightenment. I'll get my coat. -- Cheers, Steelpillow 11:49, 31 May 2008 (UTC)

Photo of Hilbert
I'm not thrilled to see the photo of Hilbert at the top of the article. I associate photographs like that with grade school textbooks, where they are intended to distract readers from the actual content. Since the intended readership of this article is not grade school students, I don't think we need to use such photos here.

But since the article is "Hilbert space", I left the photo there pending discussion. I did remove a second photo, of von Neumann - I dispute that the reader learns anything about Hilbert spaces by seeing a photo of von Neumann, and I don't think we should routinely add photographs of people simply because they are mentioned by name and/or important.

Because there are several other, very appropriate, images in the article, I don't see that these two photographs are needed for GA status. &mdash; Carl (CBM · talk) 19:58, 25 July 2008 (UTC)


 * I agree with the removal of the von Neumann photo, and suggest moving the photo of Hilbert down to the introduction and history section. A lead image is not required for GA, but a good one would help the article progress further. So far the quantum mechanics one looks like the nicest. Geometry guy 21:10, 25 July 2008 (UTC)

An editor pointed out that WP:MOSIMAGES suggests that all articles should start with a right-aligned image unless there is a compelling reason not to do so. In this case, I think I have laid out a compelling reason not to have the picture of Hilbert in the lede. What about the picture of the harmonics of a vibrating string, from lower in the article (with a different caption)? Is there another, salient, image that could go in the lede? &mdash; Carl (CBM · talk) 21:50, 24 August 2008 (UTC)
 * Your reason is not at all compelling, seeming to be a personal distate for the style of grade school textbooks. I have demonstrated that this style of using images which might interest the reader and alleviate the off-putting appearance of a wall of text and formulae is our preferred style.  As and when you find another image, we may consider their relative position but for now I shall be reverting so that the article conforms to our style guide. Colonel Warden (talk) 22:16, 24 August 2008 (UTC)


 * As I suggested above, I put the vibrating strings image in the lede. If you insist is must be 300px, please feel free to change the size. The image of Hilbert contributes nothing to the reader's understanding of Hilbert spaces,and its inclusion in the lede makes the article appear sophomoric. I have the impression that you are pushing the MOS for its own sake while ignoring the (lack of) actual benefit of the Hilbert image in the lede. &mdash; Carl (CBM · talk) 23:56, 24 August 2008 (UTC)


 * Re Geometry guy: the electron orbitals image is also nice, and more compelling as an image. But it may not be as accessible to a naive reader as a vibrating string. &mdash; Carl (CBM · talk) 23:58, 24 August 2008 (UTC)

Image Twice?
It seems a bit strange to me for the image of overtones of a vibrating string to appear both in the lead paragraph and later in applications. Wouldn't it be better to have it just in applications? Thenub314 (talk) 08:00, 10 September 2008 (UTC)


 * Some history: The photo of Hilbert was originally in the lead, and there was a photo of von Neumann in the introduction section. (See this revision.)  This was too "grade schoolish" for some editors, who preferred to have pictures illustrating applications rather than pictures of mathematicians.  So the von Neumann photo got the chop, the Hilbert photo moved down, and the string picture copied to the lead.   siℓℓy rabbit  (  talk  ) 10:58, 10 September 2008 (UTC)

I like the use of images much better in the older edit. I do not find the use of images "grade schoolish" and I would suggest we go back. Even if we don't, having the same picture twice seems very silly. We should pick on spot and put it there. Thenub314 (talk) 11:07, 10 September 2008 (UTC)


 * As I explained above, I have no objections to images that actually inform the reader about the topic of Hilbert spaces. I do object to the use of a portrait of Hilbert in the lede. What, precisely, does the reader learn about Hilbert spaces by knowing Hilbert's appearance? Nobody has proposed an answer to that question.


 * I don't see why it's a problem to have the sameimage twice, since the first use has a simple caption and the second has a more complicated caption. I thought that the repetition might actually be interesting to the reader. Seeing the image in the lede, she may wonder what the relationship is; lower down, she will see the explanation.


 * Of course, it's simple to remove the second use, or put a different image in the lede. My only objection is to the use of portraits to identify mathematical subjects. &mdash; Carl (CBM · talk) 12:25, 10 September 2008 (UTC)


 * I suppose I should have read the thread before this. My argument would be that the photos are not pertinent to the mathematics but are part of the history of the subject.  As a personal comment, I like math articles that include photos of the Main contributors.  It reminds that mathematics is a human endeavor.  But if more the consensus is to not include unnecessary pictures that is also fine.   I had read the article for the first time in a while and I just felt that the two copies of the vibrating string didn't work well.  My gut reaction was "well if they have to keep repeating the applications, the list of applications must not be that long" even though I know full well this is not the case. Thenub314 (talk) 14:29, 10 September 2008 (UTC)


 * The photo of Hilbert is included, in the History section. I don't think anyone objects to that. &mdash; Carl (CBM · talk) 14:33, 10 September 2008 (UTC)


 * Very true. It is mainly having the same image twice I object to.  I do think the was better with pictures of two different mathematicians then the same vibrating string twice. Thenub314 (talk) 15:08, 10 September 2008 (UTC)


 * Perhaps, but I think "at least it's not the same picture" isn't a very strong argument in favor of a portrait. My main concern for pictures is that convey some information about the topic at hand to the reader. &mdash; Carl (CBM · talk) 16:20, 10 September 2008 (UTC)


 * I agree completely. I think I am obscuring my point by speaking about the portraits.  I feel that having the vibrating string picture in the lead doesn't work well.  It should be solely under applications.  If we feel the need for a picture in the lead, we should get a different one or go back to the way the pictures were.  The same picture twice I think gives the impression of there only being a few applications. My only point about the portraits is that a picture of von Neumann contains more information about the history then a second copy of the vibrating string picture contains about the mathematics (In my opinion of course).  Thenub314 (talk) 06:39, 11 September 2008 (UTC)

Tensor product of Hilbert spaces
I have gathered together some material in the article Tensor product of Hilbert spaces, including the text recently added to this article, because it discusses a different way of realizing the tensor product in terms of Hilbert-Schmidt operators. siℓℓy rabbit (  talk  ) 13:51, 30 November 2008 (UTC)

Where is the linear side of the inner product?
I saw that the linear side of the (complex) inner product is the left side in the examples like Cn, but it is on the right for the representation theorem. I know it is a difficult choice! (physics or maths?) but it has to be done. Bdmy (talk) 21:04, 30 November 2008 (UTC)


 * Our inner products should always be linear in the left argument. (I guess that's the math convention as opposed to the physics one.)   siℓℓy rabbit  (  talk  ) 21:09, 30 November 2008 (UTC)


 * Very good. But the "reflexivity" section links to the "bra-ket" article, where the convention is the opposite, I am affraid. Maybe a warning,
 * In physics the notation $$ \langle x | y \rangle $$ is used, linear in y, see Bra-ket notation?
 * Several good books (Rudin, Functional Analysis for instance) use X&lowast; for the continuous dual, as there is little risk that the algebraic dual would play any serious role in the discussion. In the article here, there is (at least) one occurence of $$H'$$ for continuous dual. In the HTML version I find it difficult to see the "prime". Is there a definite policy for the article? Bdmy (talk) 11:20, 1 December 2008 (UTC)
 * A warning, if done properly, seems suitable. Using &lowast; for the dual also seems reasonable, although I may have been the one responsible for the prime in the first place.   siℓℓy rabbit  (  talk  ) 12:12, 1 December 2008 (UTC)


 * Could it be reasonable to rename the subsection "reflexivity" to something like "dual, reflexivity" as there seems to be no proper subsection about the dual, and in particular about dual being also Hilbert (and perhaps it is better not to have two subsections). Bdmy (talk) 13:12, 1 December 2008 (UTC)

Definitions and Examples
The first sentence makes no sense to me. The equation defines nothing because you have neither defined the norm nor the inner product at this point. The inner product thingy <, > in the text is useless. Why not just complete the sentence by saying the inner product . Then you may say, if necessary, that a norm can be defined using . Then say that the norm can be used to make the whole thing a metric space, if indeed that is true.

I think the first sentence should indicate the exact nature of the difference between a Hilbert space and an inner product space. I looked back at the definition of inner product space and it is very clear, that is, it clearly states what you need to add to vector space to get an inner product space. This section pretty much obscures the entire notion of the Hilbert space by insufficiently describing the differences with inner product spaces and by detailing dissenting opinions too soon.

71.220.212.120 (talk) 06:02, 20 May 2009 (UTC)

Subspaces and Linear Manifolds
Some authors, e.g., Halmos in "Intro. to Hilbert Space" and use "linear manifold" to refer to a vector subspace and just "subspace" to refer to a closed vector space. Presently, the Affine_transformation page has an incorrect definition of this.

I've already suggested in Talk:Affine_transformation that the redirect for linear manifold point here. I also think this article should have the definition of linear manifold. Thoughts?

Bradweir (talk) 01:50, 26 August 2009 (UTC)


 * A better redirect would be to vector subspace. The terminology has very little to do with Hilbert spaces.  Sławomir Biały (talk) 21:05, 26 August 2009 (UTC)


 * I have moved some content around. Linear manifold now redirects to affine space, which includes a section on subspaces.  Sławomir Biały (talk) 22:08, 26 August 2009 (UTC)


 * I agree that it's a vector space idea and doesn't need the extra structure of the inner product. However, Halmos defines linear manifold as a linear subspace, not an affine subspace.  Do you have a reference where someone uses it as an affine subspace?  Also, I think the definition as a linear subspace is better linguistically, since it's not called an affine manifold.  Maybe we should move this discussion to the affine space talk? --Bradweir (talk) 04:18, 28 August 2009 (UTC)


 * I don't have a strong opinion about it. Both uses can be found, mostly in historical sources.  The Springer Encyclopedia of Mathematics uses the term "linear manifold" to refer to an affine subspace, and I have seen this usage in other places as well.  I'd be fine with getting rid of the term altogether, as it seems more likely to confuse than inform, and is not a term in wide use.  Sławomir Biały (talk) 12:12, 28 August 2009 (UTC)


 * The EoM link was enough to convince me. Maybe just a footnote on it that some author(s) use it as a linear subspace.  I was of the impression that although his books might be a little old now (still not as old as Whittaker & Watson) Halmos was still widely used by grad students learning measure theory and functional analysis.  At least I did. Bradweir (talk) 22:37, 28 August 2009 (UTC)

Definitions
"A Hilbert space H is a real or complex inner product space..." — But, if the real case is admitted, then we should require the inner product to be real-valued in this case. Boris Tsirelson (talk) 20:57, 27 August 2009 (UTC)

"This topology is locally convex and, more significantly, uniformly convex..." — is it standard enough, to say that a topology is uniformly convex? Boris Tsirelson (talk) 21:09, 27 August 2009 (UTC)


 * Fixed. Sławomir Biały (talk) 22:13, 27 August 2009 (UTC)

Angle brackets
Thanks to User:LutzL now I see the angle brackets on my browser; before, I did not. Boris Tsirelson (talk) 11:33, 28 August 2009 (UTC)
 * Just for documentation: In HTML4 the entities lang and rang (&lang; &rang;) are translated to some unicode characters like U+2329 U+232a of the misc technical block or, depending on the browser and probably locale, to the vertical alignment charactes U+3008/9 from the CJK block that are described as "canonically equivalent". Someone must have taken those to replace the html-entities. There now exists a mathematical block with U+27E8 and U+27E9, see also brackets. There is a discussion on how to change things for HTML5 (see whatwg 2007), which still can take some time.--LutzL (talk) 09:40, 29 August 2009 (UTC)
 * I'm not sure what you just said. Is it safe to use lang and rang or not?  Oh, by the way, thanks for fixing the problem.  I got the angle brackets by copying them out of the Table of mathematical symbols, which I see you have also corrected. Sławomir Biały (talk) 12:08, 29 August 2009 (UTC)
 * It is save, they will most likely be represented by the "misc tech" brackets, that seem to be a bit thin, not the "misc math A" characters. However, there seems to exist a bot or a user group that regularly changes HTML-entities to the corresponding UTF8-characters, even if that correspondence is not optimal. So it would be better to have the correct symbols from the beginning.--LutzL (talk) 23:26, 30 August 2009 (UTC)

Merge orthogonal complement to best approximation?
This makes sense since orthogonality and best approximation are essentially the same thing. On the other hand, this would make it harder to discuss a complemented subspace problem: i.e., that every subspace is complemented is essentially a characterization of a Hilbert space. I think this is an important fact to mention somehow. I'm not sure if this is worth discussing. (The succinctness is a virtue too). -- Taku (talk) 17:42, 29 August 2009 (UTC)


 * I had the same thought about merging initially, but the results of the Best approximation section are used in the section on Duality that follows it, so it is probably best to leave it where it is for now. Something we can try for is to expand the "Orthogonal complements and projections" section, within reasonable limits.  Your characterization of Hilbert spaces is an excellent idea; in fact, this theorem has a name, doesn't it? Sławomir Biały (talk) 17:54, 29 August 2009 (UTC)

I agree. The organization of the article seems almost optimal. (Though I made few edits, I really don't see much more rooms for improvement.) About the subspace characterization. First, yes, I completely missed that a one-to-one correspondence between the sets of the same carnality is trivial. If I remember correctly, the correspondence preserves inclusion in the sense V is a subspace of W iff P_V \le P_W. Anyway, it is very natural and so it makes sense to use word "correspondence". As for the theorem, I don't remember the name nor where it can be found. I'm guessing Yoshida is too old for this :) Maybe someone can provide a ref. -- Taku (talk) 18:09, 29 August 2009 (UTC)


 * It's due to Lindenstrauss and Tzafriri. Sławomir Biały (talk) 18:12, 29 August 2009 (UTC)


 * Are you sure? Maybe Kakutani 1939? See a review of Balakrishnan, Applied functional analysis. Boris Tsirelson (talk) 18:34, 29 August 2009 (UTC)


 * The review you just quoted seems to credit Lindenstrauss and Tzafriri as well, but you are of course much more the expert on such matters than I. At any rate, this can probably be settled by checking the primary sources.  Thanks for the link, Sławomir Biały (talk) 18:48, 29 August 2009 (UTC)


 * No, my expertise is silent here; I just do not remember. However, looking at the review, I understand it as follows: the Kakutani result characterizes the norm of the Hilbert space, while the Lindenstrauss-Tzafriri result characterizes the topology of the Hilbert space. And of course, assumptions differ: in the former case, projections of norm 1 must exist, while in the latter case, continuous projections (of any finite norm) must exist. The former result is much more elementary formulated, therefore, can be understood by much wider set of readers. The latter result is much deeper. By the way, the former result matters in finite dimensions (except for 2); the latter result is void in finite dimensions. Boris Tsirelson (talk) 20:30, 29 August 2009 (UTC)


 * I see. I missed the significance of the Kakutani result in my earlier gloss, and clearly both are relevant here. Sławomir Biały (talk) 22:01, 29 August 2009 (UTC)


 * "A Banach space X is topologically and linearly isomorphic to a Hilbert space if and only if, to every closed subspace V, there is a closed subspace W such that X is isomorphic to $$V\oplus W$$." I bother: the reader may interpret "X is isomorphic to $$V\oplus W$$" as existence of some isomorphism between them. Boris Tsirelson (talk) 20:52, 30 August 2009 (UTC)


 * I tried to fix the above. I have another (small) problem. The article now says: this is a characterization of the "topology" of a Hilbert space. It is I think misleading, as all separable Banach spaces are homeomorphic (due to Bessaga I believe). One should somehow mention that we are in the category of Banach spaces with bounded linear operators as morphisms (linear isomorphic classification?). --Bdmy (talk) 21:19, 30 August 2009 (UTC)


 * Thanks both of you. This is what I had intended by the statement.  To address the latter issue, perhaps "topology" should be replaced by something like "...structure as a topological vector space..."  Sławomir Biały (talk) 21:38, 30 August 2009 (UTC)


 * I've gone ahead and made this change. Sławomir Biały (talk) 23:55, 30 August 2009 (UTC)

Hodge theory
Does anyone think Hodge theory is an application worth discussing in the article? A Hilbert space, namely L^2 space plays a key role in the theory (at least a part that deals with compact manifolds). It also gives a nice example of best approximation in a function space. -- 18:14, 29 August 2009 (UTC)


 * Hodge theory is mentioned already, but in somewhat of an offhand way. I suspect that giving a proper treatment involves going too far beyond the scope of the article. Sławomir Biały (talk) 18:50, 29 August 2009 (UTC)