Talk:Hilbert space

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Apparently I'm not the only one who thinks this article is essentially incomprehensible. I added the template {{Technical (expert)}} 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)

[edit] 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 × V → 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 Wikipedia:Guide to writing better articles#First sentence), 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
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. — 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.) — Tobias Bergemann 22:39, 9 March 2007 (UTC)
Well done!
If you'll accept
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. — 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 {{technical (expert)}} 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)

[edit] 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)

[edit] 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.
\bold{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)
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