Talk:Hilbert space

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Hilbert space (reviewed version) has been listed as a good article under the good-article criteria. If you can improve it further, please do. If it no longer meets these criteria, you can delist it, or ask for a review.

	This article is within the scope of WikiProject Mathematics.
	Mathematics grading:	GA Class	Mid Importance	Field: Algebra
Add a "Further reading" appendix with a list of basic references.

Contents 1 Old talk 2 von Neumann 3 Sarfatti's defense of his Hilbert space article 4 The Layperson 5 this is wrong 6 Rewording "norm of differences" 7 frowning on bra-kets 8 What does this mean? 9 Hilbert Spaces and Topology 10 Definition section revision request 11 Good Article nomination 12 Definition

[edit] Old talk

The section "bases" is reproduced verbatim in the article orthonormal basis. I assume this is done on purpose, but I don't see the reason for it. I do see a danger however: improvements in this article may not be incorperated in the original, and vice versa.

In the same section, I don't understand the sentence "only countably many terms in this sum [referring to the Fourier series] will be non-zero, and the expression is therefore well-defined". I suppose "well-defined" means that the sum is finite, but this does not follow immediately from the fact that the number of non-zero summands is countable. Am I misunderstanding the sentence? -- Jitse Niesen 15:17, 1 Mar 2004 (UTC)

for a generic vector space, only finite sums of vectors make sense. if the vector space has a norm, and is complete with respect to this norm, than you can take countably infinite sums of vectors, and use the norm to define a limit of this sum. if the partial sums form a Cauchy sequence, then this series is guaranteed to converge to a vector in the vector space (completeness). so countably infinite summations make sense, and might converge. none of this applies to uncountable summations.

---

This article seems to assume a great deal more knowledge in the reader than can reasonably be expected. Obviously, when you're dealing with abstract mathematical concepts, it gets difficult to explain things in general terms, and without referencing other abstruse concepts and vocabulary. Still, I think Wikipedia can do a lot better than this (and it has-- see: Quantum Mechanics). After reading the article, I still had almost no idea of what a Hilbert space actually is. I can't imagine anyone who hasn't studied higher mathematics getting any use out of the article in its current form.

Thankfully, I managed to find a satisfactory explanation here. I think this too might be beyond what an average person can follow, but it's a good example of how to explain a mathematical concept in terms of physical phenomonon, and without reverting to mathematicianese. It also gives much more background to the Hilbert space than this article affords it, and which the subject deserves.

That page was written by Jack Sarfatti, a known crackpot. Many parts of his explanation are bad, in my opinion. Some parts are almost wrong, like this one, for example: "Hamilton formulated a new description of classical mechanics which was eventually housed in an infinite-dimensional phase space. In this space, a point represents the entire physical system." I would suggest not relying too heavily on Sarfatti's writings.

as far as this page goes, well, I suppose a certain amount of linear algebra and analysis are prerequisites for understanding Hilbert spaces. Perhaps it is possible to write to a less experienced audience, but I don't think that Sarfatti provides a good model for how to do so. Lethe

To understand Hilbert spaces, undoubtedly, but to explain what they are, what they are used for, and why they are significant, in a way that most people can follow? I don't doubt the mathematical explanations in this article are complete and accurate, and they should remain for those that can understand them. But encyclopedias need to be written primary for a lay-audience, not math majors. Where Sarfatti succeeds, regardless of the veracity of his article, is in framing the subject in a general context, and elucidating it in terms and analogies that are reasonably accessible. Wikipedia should strive for those qualities. I would love to be able to help improve the article myself, but obviously I don't understand the subject terribly well, so I must leave that job to those who do.

I'm not sure that I agree with you that all encyclopedia articles should be aimed at lay people. For example, the encyclopedia needs to have an article on the monster group, but I don't think there is any sensible way to explain what the monster group is to someone who doesn't know some group theory. Can you convince me that monster group should be aimed at the lay person? or how about Kähler manifold? that article is necessarily only readable by somewhat more mathematically experienced reader. In these cases, it is probably not even worth trying to aim the article to a layman, since the layman would most likely have no interest in these subjects.

The hilbert space article is somewhat similar, there are prerequisites to knowing a hilbert space that are simply unavoidable. Nevertheless, if you can suggest which parts are hard to follow, and why, I might try to help. Lethe | Talk

Hilbert spaces are different because they are linked from a lot of articles on quantum mechanics, making its relations to quantum mechanics more understandable to people who might have an interest in quantum mechanics but who perhaps only know basic calculus and linear algebra, like many chemists for example, would perhaps not be a bad idea. Passw0rd

I also agree that this article needs to be of more use to a lay person. This is a constant problem with mathematics and physics related articles on wikipedia. I'm not really sure what the point of writing an article that is indecipherable to a non-expert is, in this context; I would imagine that anybody who could understand this page, already knows what a Hilbert space is, and is certainly not going to be looking for an explanation of it on wikipedia. Sayfadeen

It seems if we are going to talk about Hilbert spaces in relation to the QM, then we should, at some point, include a link to "Rigged Hilbert Space" since that is what is actually referred to in QM.

[edit] von Neumann

As I found it, the page on Hilbert spaces was incorrect to speculate that the abstract spaces were invented by Weyl in 1931. I have corrected the page to cite the 1929 paper in which von Neumann coined the term.

[edit] Sarfatti's defense of his Hilbert space article

It's one thing to say my article is not "good", it's quite another to say I am a "crackpot". I challenge Lethe to give even ONE specific example in which I wrote something about physics that is "crackpot". Making errors as all do is not same as being "crackpot". I use only mainstream physics. I do not say that relativity is wrong. I use relativity. Ditto with quantum theory. So what is Lethe talking about? BTW:

I didn't claim that you were a crackpot because the article was wrong. I claimed that you were a crackpot and the article was wrong. The implication was that if you are a crackpot, of course your articles about Hilbert spaces will be wrong. -Lethe | Talk 02:20, 28 October 2005 (UTC)

Do you mean "Hamilton formulated a new description of classical mechanics which was eventually housed in an infinite-dimensional phase space. In this space, a point represents the entire physical system."

What's wrong with that? I am not saying phase space is Hilbert space. Is that what you assumed? In any case that statement hardly makes me a "crackpot". The phase space of classical fields is infinite.

These are the mistakes in that sentence, as I see them. First, Hamilton never formulated classical mechanics in infinite dimensional phase space, nor did his description eventually become housed in infinite dimensions. Hamiltonian mechanics is formulated in phase space, which is 6n dimensional for n particles in 3 dimensional space. This is finite. Second, in both quantum mechanics and classical mechanics, finite dimensional or not, a point in the phase space represents only one possible state of a physical system, not the entire system. The physical system is capable of other configurations, which are not represented by a point in phase space. -Lethe | Talk 02:20, 28 October 2005 (UTC)

Looking at Lethe's page he seems to be a mathematician with the typical arrogance mathematicians have toward theorertical physicists and vice versa. Feynman referred to their penchant for "rigor mortis" that is evident in Lethe's remarks here. I have a PhD in physics from the University of California BTW. Lethe seems to be part of John Baez's clique trying to take over the Internet Math/Physics which explains his "crackpot" remark since John Baez has held a personal grudge against me for more than ten years including spreading a false story about me and Gell-Mann, which never happened. What Baez did do was to garble a story Ed Siegel told me about him and Gell-Mann. Baez simply substituted my name where Ed Siegel's should have been. Jack Sarfatti JackSarfatti 00:12, 28 October 2005 (UTC)!

I was going to complain about ad hominem attacks, but I suppose I started this whole debate with an ad hominem against you in the first place, by calling you a crackpot. So go ahead, attack my credentials and my motives. -Lethe | Talk 02:20, 28 October 2005 (UTC)

Whoa! OK, I understand you don't like being called a crackpot. But Lethe made that comment over a year ago, and I think some sort of statute of limitations would apply to off-the-cuff remarks. And I'm one of those arrogant mathematicians, so please I feel offended. Give a link to your side of the story about Baez. That would help.--CSTAR 13:02, 28 October 2005 (UTC)

A little googling turned this up, though I was in grammar school when that was written, so I'm not sure what it has to do with me (other than that I'm a fan of Baez' This Week's Finds). -Lethe | Talk 00:55, 5 November 2005 (UTC)

[edit] The Layperson

I've noticed that nearly all articles of a scientific or mathematical nature on wikipedia are nearly indecipherable to the kind of person who would most likely be referencing the subject on wikipedia. While I'm sure the article is well-researched, I doubt that the kind of person who could decipher most of what's in this page would be looking it up on Wikipedia.

Now, that being said, it is true that part of the wikipedia concept is that it should be a storehouse of knowledge. I'm therefore not suggesting that anything in this or other articles of a similar nature should be deleted; as a "record" of knowledge it seems to stand pat (though I'll let the experts dispute that if they want, I have no idea myself, I'm just speaking in terms of content style); but there should be an effort to include a seperate "layperson's" section in these types of articles that explains the basic concepts to the person most likely looking up this type of article. In this case it is most likely somebody who is trying to learn about qunatum physics and has stumbled across the term "hilbert space", has no idea what it means, and goes to wikipedia to unsuccesfully find clarity.

And, no, I'm not volunteering, because I'm a layperson on this subject myself. I've certainly endeavoured to do this in my area of speciality. If somebody wants to test their teaching skills, I'd love for somebody to decrypt these types of things. (Unsigned comment from User:Sayfadeen 17 January 2006).

Yes, this article starts out a bit too abstract, and should provide a simple example, probably before the formal definition. In the meantime, the best two simple examples are the discrete Fourier transform, and the orthogonal polynomials. Once you understand those, perhaps this article will become accessible. linas 06:06, 18 January 2006 (UTC)

I got no clue what a hilbert space is by reading the article, i'm not a scientist. I've got some math at school but not this much. Isn't there a simple kind of figure who might explain this 'space' thing since it's called space i asume it's a kind of shape. Perhapps with some strange future's. I'm just like the 95% of the world who believes a picture can tell more then 1000 words. pgt2006 2 feb 2006

It's pretty hard to draw pictures of things that are 4-dimensional. It's inconceivable to draw pictures of things that are infinite dimensional. -lethe ^{talk +} 21:15, 2 February 2006 (UTC)

Listen you guys, it's hard enough to write a math article that's correct, it's harder still to make it correct, complete, and still as comprehensible to the non-expert as it can be. It's quite a task! While no one wants to make their articles inaccessible, it's certainly incorrect to say that experts do not use Wikipedia. We serve experts and nonexperts alike here. Thus, completeness and correctness are always important goals, and they're probably easier to attain. We serve nonexperts as well, so that accessibility is an important goal as well, though it might be harder. Which goals are more important is the subject of various debate, suffice to say, this is well-worn territory, so you don't have to theorize about the relevance of super-technical material on wikipedia. Let's focus much more specifically: this article assumes too much prior knowledge right from the first sentence, and needs help. We can certainly do that, let's give it a try. Now, I've just rewritten the intro paragraph. It would be helpful if you would comment on the new intro. Also, perhaps you could describe exactly which parts are too technical to your eye. The technical matter can't be forgotten, but it can certainly be postponed and contextualized. My point is, instead of complaining about how inaccessible it is, help us to improve it. Even if you don't feel you're qualified to rewrite it yourselves, you can help just by saying which words are too technical too soon. This is a wiki, afterall! -lethe ^{talk +} 21:34, 2 February 2006 (UTC)

Go lethe! linas 00:59, 3 February 2006 (UTC)

I am a physicist and i have started to use wikipedia cause it has grown to a point where it is quite useful. Its a convenient way to remind me of basic things that I may have forgortten, or didnt learn that well in the first place. Thanks to you-all! Physicists use Hilbert spaces and other n-dimensional spaces to make it easier for them to *mathematically* manipulate and describe physical phenomena to other physicists. They are tools of mathematical algebraic convenience and impossible to describe in a 3 dimensional euclidean space picture. As such, they are of no use and simply confusing for lay people.69.44.253.177 01:46, 14 August 2006 (UTC)

[edit] this is wrong

"If a linear operator is defined on all of a Hilbert space then it is necessarily bounded. However, if we allow ourselves to define a linear map that is defined on a proper subspace of the Hilbert space, then we can obtain unbounded operators."

I've deleted this, http://en.wikipedia.org/wiki/Discontinuous_linear_map gives a general example of unbounded operator on any infinite dimensional normed space into any non-zero normed space, so this is not true. —The preceding unsigned comment was added by Scineram (talk • contribs) .

That should have read "if a closed linear operator is defined" etc. I'll fix it.--CSTAR 14:57, 27 February 2006 (UTC)

[edit] Rewording "norm of differences"

"Completeness in this context means that every Cauchy sequence of elements of the space converges to an element in the space, in the sense that the norm of differences approaches zero."

This confused me for a while until I realized "norm of differences" meant "difference of the norms" (or at least I think that's what it means). Based on my background knowledge of a Cauchy sequences, I think we mean to say that any sequence of vectors in Hilbert space H that has the property such that the sequence of norms corresponding to those vectors is Cauchy will converge to the norm of a vector in H.

I.e. v1, v2, v3, v4... will have corresponding norms: n1, n2, n3, n4.... Since the norms are scalar numbers, we define them being Cauchy in the usual way, and the sequence of norms: n1, n2, n3, n4... must converge to the norm of some vector in H.

If I have guessed right, I move to change "norm of differences" to "difference of norms" or even "difference of the norm of the vector being approached and the norms of the vectors in the sequence", as the article doesn't mention what two things the difference is of.Loodog 01:15, 7 March 2006 (UTC)

You have not guessed right. The criterion for Cauchy convergence of a sequence {x_i}_i is that ||x_i–x_j|| get small. This is a norm of differences. If instead we asked that the difference of norms got very small, what would happen with an infinite sequence of independent unit vectors? All their norms are 1, so all differences of norms are 0, but this sequence does not converge (and so should not be Cauchy). -lethe ^{talk +} 01:24, 7 March 2006 (UTC)

Very true, but how is the difference of two Hilbert space vectors defined? Loodog 03:48, 7 March 2006 (UTC)

--CSTAR 03:52, 7 March 2006 (UTC)

I guess the problem is I'm trying to contexualize this in L2. What would the difference of f(x)=x^(3/2) and g(x)=exp(-x^2) be?Loodog 04:07, 7 March 2006 (UTC)

Well, I'm not sure what your domain is, if it's L²(R), then the first function is no good: it's not square integrable. If your domain is something smaller, [0,1] for example, then you should be OK. Anyway, in any function space, sums, differences and scalar multiples are defined pointwise, so the difference is just x^(3/2) – exp(–x^2). -lethe ^{talk +} 04:14, 7 March 2006 (UTC)

[edit] frowning on bra-kets

Small Question - Can anyone show me an example of a situation in which "bra-ket notation . . . is frowned upon my mathematicians". While I am in the dept of physics, I've never heard any of the math-folks say a single bad thing about. Granted, they wouldn't use it to do proofs or anything but they have all acknowledged it's amazing utility.Wrath0fb0b 17:07, 9 March 2006 (UTC)

I guess one of the problems with bra-ket notation is that you often write a ket with an eigenvalue as its label, and for some operators, there are points in its spectrum for which there are no vectors in Hilbert space. So over-reliance on eigenvector-eigenvalue correspondence is one problem. I guess the other is that it obscures the distinction between a Hilbert space and its dual. Even though it's OK to do so because of the Riesz lemma, I think mathematicians prefer to maintain the distinction. -lethe ^{talk +} 17:21, 9 March 2006 (UTC)

Off-topic, maybe, but .. coming from a physics background, I eventually learned that not all operators are trace class, and this was a shock and a big conceptual problem I'm still wrestling with. Worse, one actually does encounter non-trace-class operators in physics. There's a certain naivete in physics that can get you into trouble when one starts getting into harder material. linas 01:54, 10 March 2006 (UTC)

[edit] What does this mean?

"Since all infinite-dimensional separable Hilbert spaces are isomorphic"

Isomorphic as what? And are separable and inseparable spaces isomorphic? —The preceding unsigned comment was added by 59.144.16.174 (talk • contribs) .

Isomorphic as Hilbert spaces. An isomorphism of Hilbert spaces is an invertible linear isometry. A separable Hilbert space is not isomorphic to a nonseparable Hilbert space. This is the Hilbert space analogue of the statement from linear algebra that two vector spaces are isomorphic if and only if they have the same dimension. -lethe ^{talk +} 08:07, 24 April 2006 (UTC)

[edit] Hilbert Spaces and Topology

I am reading Alain Connes' Noncommutative Geometry, and in the first chapter he writes an innocent sounding statement (paraphrased): up to isomorphism, there is only one Hilbert Space with a countable basis.

"Countable basis" is a topological term, and the "basis" used here appears to be the linear algebra context. There does not appear to be a mention of the topological underlying of a Hilbert space on this page. I doubt its insignificant. Should that be added? I would do it, but I came here to find the answer to my question so I'm obviously not the best person. --68.98.221.241 22:58, 30 June 2006 (UTC)

I have heard that statement often. It's true, but pretty stupid. Anyway, I'll try to add something about the topology to the article. -lethe ^{talk +} 23:24, 30 June 2006 (UTC)

Well I added some stuff to the article which might help. Let me know if it does. -lethe ^{talk +} 23:53, 30 June 2006 (UTC)

[edit] Definition section revision request

Could someone please revise the Definition section of this article in the following ways:

Set the actual definition off from the introductory material (probably a separate paragraph; perhaps without a pronoun reference to 'this norm').

Clarify the distinction between a Hilbert Space and a Banach Space. Both articles mention a (cauchy convergence) complete normed space. The Hilbert Space Definition mentions that inner products give rise to a norm but doesn't make it clear and explicit that the norm must be the inner product norm that is mentioned at the top.

Move the examples of the space to the examples section.

--DrEricH 23:09, 21 August 2006 (UTC)

[edit] Good Article nomination

Firstly, I would like to mention that I am not a mathematician. The last time I covered Hilbert spaces was in an undergraduate Linear Algebra course over 20 years ago and I only understood the concept in the context of quantum mechanics. Therefore I will not comment on content, but on editorial standards and style. In my opinion, this is a Good Article, however, I would like to make a few suggestions for improvement:

• In the mathematics Manual of Style, it is suggested to include “an informal introduction to the topic, without rigor, suitable for a high school student or a first-year undergraduate”. Perhaps just one or two sentences is required in the introduction.
• The applications of Hilbert spaces (quantum mechanics, PDEs, stochastic processes) should be in the “Introduction” section rather than the “Definition” section.
• It is mentioned that Dirac notation is frowned upon by mathematicians. Having “grown up” on the notation (in the field of physics), I was surprised to read that. If it is stated in the article here, for completeness, it may be wise to state why that is the case. The article on Dirac notation unfortunately doesn’t mention why (probably because it was written by physicists).
• The section “Operations on Hilbert spaces” has only one sentence. Can this be put into another section or expanded?
• As noted in the mathematics Manual of Style, the use of the first person (“we”) should be avoided.
• I would suggest giving a list of basic references, under a “Further reading” appendix section.

I have placed the article “On hold” for the changes to be implemented. RelHistBuff 10:03, 10 October 2006 (UTC)

I tried to implement most of these suggestions and make some other basic changes. I think the page could still be improved quite easily. I didn't make too much progress on the informal introduction and didn't add any basic references. I did try to clean up the language a little bit and remove some unnecessary chaff. –Joke 15:01, 10 October 2006 (UTC)

The statement that the bra-ket is "frowned upon" by mathematicians seems a little histrionic. Just because something is not used doesn't make it "frowned upon." Moreover, in a couple of places, it was suggested that infinite dimensional Hilbert spaces are the only ones that matter. That is definitely not true: many authors use Hilbert spaces when they could, and often do, mean finite dimensional spaces so that they can treat the finite and infinite dimensional cases together. –Joke 15:03, 10 October 2006 (UTC)

The article looks better now. For a basic reference suggestion, I thought about the textbook I used, but I can't recall the title and in any case it's probably out-of-print. A short search turned up this possibility: Theory of Linear Operators in Hilbert Space by Akhiezer and Glazman. The reason I thought this one might be good is that it is a Dover publication, one of my favourite publishers for good, basic, low-cost, and classic math and physics textbooks. Probably the experts here know of better or more examples. RelHistBuff 09:40, 11 October 2006 (UTC)

I came by again and although it is still a day early from the hold limit, I promoted the article to GA. I would still recommend adding a "Further reading" appendix with a list of basic references. However that does not stand in the way that this is a good article. RelHistBuff 07:42, 16 October 2006 (UTC)

[edit] Definition

I wanted to suggest maybe opening the section on the definition slightly differently. Something like "A Hilbert space is a vector space with some additional stucture. In particular it has an inner product and it is complete with respect to topology created by this inner product. To understand what this means notice that every inner product....." I am not sure what I have written is any good but I feel it could use a bit of a description of where we are going before we start speaking of open balls and topology. Any thoughts Thenub314 12:44, 11 October 2006 (UTC)