# Talk:Hilbert space

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I don't have time for a full blown GA review, but I do have some comments that may need to be addressed before GA is passed.

• My first comment is about the accessibility of the lead. I fear that you will have lost most of your potential readership by the end of the first paragraph. Most engineers and physicists do not know what a Cauchy sequence is, and will only have a very vague notion of the concept complete. To make the article accessible without a background in mathematics, the first paragraph needs to explain complete in more every day terms. One option (that makes me and probably every mathematician here cringe): "... is complete, meaning that if every converging sequence of vectors has a limit within the space." Of course, using the term converging in this way is not really mathematically correct, but it does a much better job at transferring the idea at an intuitive level. But I will be the first to concede that this option also isn't optimal. There must be a better solution.
• The article claims that almost all Hilbert spaces in physics are separable. This is not true pretty much all Hilbert spaces occuring in quantum field theories (= most of modern theoretical physics) are not separable. (They are generated by an orthonormal basis |p1,p2,...>, where the pi's are the momenta and thus continuous parameters.)
• The article makes frequent use of ⟨ and, which do not render on my work PC, Which runs a quite standard XP install. Is it possible to find an option that works on more PCs?

Maybe I'll comeback for a more detailed pass later. (TimothyRias (talk) 08:26, 8 September 2009 (UTC))

Another option: "... is complete, meaning that it is not a dense subspace of a larger space." Boris Tsirelson (talk) 10:48, 8 September 2009 (UTC)
About nonseparable spaces: are you sure? The usual representation of a free field (be it bosonic or fermionic) is in the separable Fock space. Or do you mean something essentially non-free? As far as I understand, a physicist can write |p1,p2,...> not because he really needs nonseparability, but rather because on his, somewhat heuristic level, it is usual to treat the continuum as just a very fine lattice. Boris Tsirelson (talk) 10:53, 8 September 2009 (UTC)
Well, supposedly the |p1,p2,...> are supposed to be orthogonal to each other of the pi's don't match. So, unless you know a way to make an uncountably infinite number of mutually orthogonal vectors in a separable Hilbert space this means that the Hilbert space must be non-separable. But there might be way to formulate it more carefully in which the Hilbert spaces are separable. (If space is assumed to be compact for example.) My understand of this has always been that physicist simply don't care. (TimothyRias (talk) 11:50, 8 September 2009 (UTC))
The old version of the lead had "complete, meaning that if a sequence of vectors approaches a limit, then the limit is guaranteed to be in the space as well". I will try this: "complete, meaning that if infinite summation of vectors converges absolutely, then it converges to some limit within the space." This has the advantage of being technically accurate, and probably communicates more to engineers than the current version. Sławomir Biały (talk) 11:04, 8 September 2009 (UTC)
I think that could work. To make it even more accessible you could also spell out converges absolutely to something like "... if the norm of an infinite summation of vectors converges than it must have a limit within the space." Or something along those lines. There may be a significant subset of potential readers that wouldn't know what absolute convergence is. (But does know what a series, limit and converging mean.) Anyway, your proposal already reaches many more readers than just "Cauchy". (TimothyRias (talk) 11:50, 8 September 2009 (UTC))
We usually teach absolute convergence in freshman calculus, so I think it is reasonable to expect that most potential readers will have at least some familiarity with the concept, although probably not for vectors, and certainly not in infinite dimensions. I would also resist trying to spell things out too much. Too much detail is just as likely to make the lead inaccessible as too little. The lead could say "converges absolutely (in norm (mathematics))" instead of just "converges absolutely", at least to give the term "absolute convergence" some more meaningful context. However, the term "norm" is more likely to be unfamiliar than "absolute convergence", I reckon, and taking the time to spell it out is likely to make the description of the space too long. Less is more. Sławomir Biały (talk) 12:17, 8 September 2009 (UTC)
Although absolute convergence is usually taught in most physics bachelors it is one of those subject which most (physics) students ended up forgetting again. (simply because it is a concept they rarely use.) Vectors on the other hand are the bread and butter of many applications and are thus likely to remain fresh in mind. I agree however that the lead should also not expand too much in fear of becoming unreadable. Your suggestion of the (in norm) addition will probably do the trick for most people. (TimothyRias (talk) 12:32, 8 September 2009 (UTC))

I have added a paragraph with a reference to the Separable spaces section to address your second comment. Sławomir Biały (talk) 12:19, 8 September 2009 (UTC)

That seems to cover it. Good work! (TimothyRias (talk) 12:32, 8 September 2009 (UTC))

As for me, your remarks about nonseparability in QFT do not conform to the fact that the Fock space is separable (and your arguments do not involve non-Fock spaces). Boris Tsirelson (talk) 13:58, 8 September 2009 (UTC)

Actually, Fock spaces have little to do with it. The Hilbert space of single particle state in QFT is already nonseparable. Considering multiple paticle states actually makes this even worse. This simply comes down to the fact that [a(p),a(p')]=(2π)3δ3(p-p') => |p> and |p'> are orthogonal if p!=p'.(TimothyRias (talk) 12:03, 9 September 2009 (UTC))
No. In the same way you could say that the Hilbert space of the quantum mechanics is nonseparable. Indeed, it is rather usual (among physicists, not mathematicians) to use spaces |x> concentrated at points x. Of course, their scalar product is also a delta-function, and so, they are orthogonal. However, all that is not about separability, it is about a physicist's way to use delta-function as if it was a function. In fact, the state |x> is non-physical in many aspects: it is of infinite moment, of infinite energy etc. (Not that the mean energy is infinite, but much worse: energy is infinite with probability one.) Physical states are square integrable functions. Square root of delta-function is not. And the same holds in the quantum field theory. At least if you do not leave the Fock space. Physicists often say they do leave, but when they are serious about it, they mean another separable space. Boris Tsirelson (talk) 15:26, 8 September 2009 (UTC)
Also, Timothy, do not forget to sign... Boris Tsirelson (talk) 15:30, 8 September 2009 (UTC)
Let me add: it is common knowledge that (at least in the framework of a quantum theory of free bosonic massive field) that the one-particle sector of QFT is the same as the Hilbert space of quantum mechanics (of a single particle). (On the level of kinematics I mean, not dynamics.) If in doubt, ask Physics project. Boris Tsirelson (talk) 15:34, 8 September 2009 (UTC)
Yes your right. I got confused by the bad habits of physicists. (TimothyRias (talk) 12:03, 9 September 2009 (UTC))
Nice. Happy editing. Boris Tsirelson (talk) 13:48, 9 September 2009 (UTC)

Some early feedback, some acronyms are introduced out of the blue (PDE for partial differential equation, L2 is dropped without proper introduction, and so on). Now I know what these are, but I also have a background in physics. Also I notice nothing is mentioned about Banach spaces and recall hearing that Hilbert & Banach spaces were intimately related (never worked with Banach spaces, so I could very well be 20 miles off track here!). Headbomb {ταλκκοντριβς – WP Physics} 17:36, 8 September 2009 (UTC)

I've gotten rid of the unexpanded acronyms. Keep a keen eye out for that sort of thing. I've thought about how best to introduce Banach spaces. Although Hilbert spaces are indeed Banach spaces (and inherit all the special properties that Banach spaces have), this is a conspicuously under-emphasized aspect of the theory. I don't know what the appropriate level of coverage is, then, but I sense that it is more than we have now but possibly less than a full section. In a section, I find myself wanting to say inane and obvious things like, "the open mapping theorem/Hahn-Banach theorem/closed graph theorem also holds in Hilbert spaces", but I don't really have anywhere to go with it. I think the best approach may be to integrate some material into the existing article here and there. For instance, the closed graph theorem should be mentioned in the section on operators and the Hahn-Banach theorem in the section on duality. Sławomir Biały (talk) 18:32, 8 September 2009 (UTC)

## Jargon

The Clarify-jargon template I added was summarily removed as a "drive-by tagging", whatever that is. So without re-adding it, let me explain why I put it there in the first place: I have no idea what Hilbert space is and I have tried to get through this article several times. Since I am a smart person who studied maths to A-level I conclude that my experience is not unrepresentative. To understand the first paragraph, you are asking readers to get a grip on: Euclidean space, vector space, complete metric space, mathematical series, absolute convergence, and norm (mathematics). It's just too much. My feeling is that you need something much more general:

A Hilbert space is an abstract spatial concept used by mathematicians to consider the relations between certain types of vectors (quantities which have both magnitude and direction). As such, it is one of a number of different kinds of vector spaces, and is named after David Hilbert, who helped develop the idea. While many other vector spaces are concerned with a two-dimensional plane, or three-dimensional space, a Hilbert space extends such ideas into an infinite number of dimensions. They are an indispensible tool in many areas of mathematics, physics, and engineering.

...with further technical details coming in gradually lower down. The whole article has similar problems, however, which is why I tagged it. Widsith (talk) 15:54, 8 September 2009 (UTC)

Nowhere in WP:JARGON, WP:MTAA, or WP:LEAD does it say that the lead should be so watered down as to say virtually nothing about the subject, as the above paragraph does. On the contrary, the lead is supposed to define and give context the subject, and to summarize the contents of the article. The current lead does this fairly thoroughly, I think. While it is perhaps not accessible at a level where someone without any mathematical background (indeed, would such a person even have a use for an article about Hilbert spaces?), it truly does not overindulge in technical details. The present first paragraph attempts to introduce Hilbert spaces in the simplest possible way, and offers up an additional sentence to say why both the inner product and completeness are important. Sławomir Biały (talk) 16:49, 8 September 2009 (UTC)
You ask what need someone without a mathematical background would have of this article. The need I have, being such a person I suppose, is that I want to know what a Hilbert space is and I expect an encyclopaedia to tell me. If you are writing this for an audience already versed in the subject, then it seems to me you've missed the point of the project. If you think this is the "simplest possible way" to introduce the subject, I can only say I don't believe you. Widsith (talk) 17:06, 8 September 2009 (UTC)
For instance, you have objected to the use of the word "completeness" in the lead. This is actually a fundamental aspect of the Hilbert space, just as important as the "linear space" aspect of the theory. So, if you want to understand Hilbert spaces, you also need to understand the implications of completeness. Rewriting the lead so as not to mention this fundamental property is a major omission, and will certainly not help anybody to understand the concept. From WP:MTAA: Do not "dumb-down" the article in order to make it more accessible. Accessibility is intended to be an improvement to the article for the benefit of the less-knowledgeable readers (who may be the largest audience), without reducing the value to more technical readers. Sławomir Biały (talk) 17:11, 8 September 2009 (UTC)

I have rewritten the first paragraph to address this concern. Sławomir Biały (talk) 18:18, 8 September 2009 (UTC)

## potentially infinitely many dimensions

"...to spaces with any number of dimensions (or coordinate axes), including potentially infinitely many dimensions..." What is meant by "potentially" here? Just "possibly", or indeed something in the spirit of Actual infinity#Aristotle's Potential-Actual Distinction? Boris Tsirelson (talk) 17:48, 9 September 2009 (UTC)

It just means possibly. Sławomir Biały (talk) 17:59, 9 September 2009 (UTC)

The author references Kolmogorov-Fomin Real Analysis, Prentice Hall (1970.) It may be confusing to some readers who know the book, that Kolmogorov defines Hilbert Spaces to be strictly infinite dimensional. On p. 155 of K&F Hilbert Space is defined as "A Euclidean Space which is complete, separable, and infinite-dimensional." Other classic references also use the Kolmogorov definition. e.g. Dennery and Krzywicki Mathematics for Physicists Dover (1995) p.197 (seems to). In many (older) references, finite dimensional Hilbert Spaces are not defined. Probably useful to readers to acknowledge the various historical usages of the term, especially so, for an encyclopedia article. A few words contrasting the various definitions would seem appropriate. BTW, nice work on this detailed article.

suggestion: "Some older references define Hilbert Spaces to be strictly infinite dimensional. e.g. Kolmogorov and Fomin (1970)." — Preceding unsigned comment added by Mathview2011 (talkcontribs) 00:58, 27 May 2011 (UTC)

Actually, many older references require that a Hilbert space be infinite dimensional and separable. We already mention the separability thing. Would it be enough just to add "infinite dimensional" to that statement, in the section on separability? Sławomir Biały (talk) 12:28, 27 May 2011 (UTC)

## Remarks to the "Quantum mechanics" section

"the state space for position and momentum states is..." — seems ugly; someone could think that a "position state" is the state localized at a point (but in fact, there is no such state in this space). Maybe , rather something like "the state space for a single nonrelativistic spinless particle moving in the three-dimensional Euclidean space"?

"while the state space for the spin of a single proton is just the product of two complex planes." — or maybe "while the state space for the spin of a single spin-1/2 particle (for example, proton) is two-dimensional (just the product of two complex planes)."

"Each observable is represented by a maximally-Hermitian (precisely: by a self-adjoint) linear operator" — I do not remember the term "maximally-Hermitian"; what is it? Also, "Hermitian" usually assumes "bounded", while "self-adjoint" does not.

"If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues." — Yes; and if it is not discrete then what? Then the observable can only attain values belonging to the spectrum.

"During a measurement, the probability that a system collapses..." — Yes, but all that is about an ideal quantum measurement (which is much simpler that the general case described via "ancillas", "effects", "quantum operations", "quantum instruments" etc).

"Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute." — Really? Does it also (or rather, mainly) treats some, quite nontrivial implications of the non-commuting? And not quite from non-commuting, but rather, from the commutant being a scalar operator.

Boris Tsirelson (talk) 12:29, 11 September 2009 (UTC)

This section systematically refers to the Hilbert space as the "state space" of quantum mechanics. Formally speaking, this is wrong. Formally, the space of states is the space of positive linear functionals on the operator algebra that send the unit operator to 1. This space is generally much larger than just the unit elements of the Hilbert space (which it naturally contains). The states that are related to Hilbert space elements are called "pure". All other states can be written as linear combinations of pure states and are called "mixed". The proper classical analogue of the Hilbert space is phase space, with the operators corresponding to functions on this phase space and state being distributions on the phase space. Now, this is mostly formally and it is not unusual for physicists to refer to elements of the Hilbert space as states, I would just hold of on calling the Hilbert space "the state space". (TimothyRias (talk) 08:53, 23 September 2009 (UTC))

Yes and no. This (above) is one of approaches. According to this approach, a physical system is described by an algebra of observables. According to another approach, a physical system is described by an algebra of observables AND ITS REPRESENTATION in a (separable) Hilbert space. Another representation of the same algebra describes another physics. The distinction between pure and mixed states is important, of course. However, there is another important distinction: between normal and non-normal states. A normal pure state corresponds to a one-dimensional subspace of the (given) Hilbert space (or equivalently, to a unit vector, with the reservation that phase factor does not matter.) A normal mixed state corresponds to a "density matrix", just a trace-one positive Hermitian operator. Non-normal states can be pure or mixed, too. Some scanty information about normal states can be found in Von Neumann algebra.
Consider for example the "usual" quantum mechanics of a one-dimensional spinless particle (non-relativistic). The "usual" representation is the irreducible representation given by the von Neumann theorem. A normal pure state corresponds to a square-integrable wave function; it cannot be concentrated at a point (of the one-dimensional "physical" space). A normal mixed state also cannot be concentrated at a point. In contrast, a non-normal state can. In fact, for a given point (of the one-dimensional "physical" space) there are a lot of states concentrated at this point; they all are non-normal; some of them are pure (still a lot). See also the discussion about separability in the section "GA comments" above. Boris Tsirelson (talk) 16:51, 24 September 2009 (UTC)
I wasn't trying to imply that that was the only formalism out there, but whatever formalism you take the physical role of the Hilbert space, it is never really just that of "state space", as it said in the article. Some more nuance is/was required on that point. (TimothyRias (talk) 10:26, 25 September 2009 (UTC))

A mathematician I used to work with, Kane Yee (best known for a numerical algorithm), said that the space of electromagnetic waves is actually only a Banach space because the evanescent and free solutions have different normalization. This appears to apply also to qm. Bound states of negative energy and free states of positive energy are normalized differently. This is "fixed" by introducing a fictitious box, making all states bound. Julian Swinger said that one actually uses four such boxes, each infinite with respect to what is inside it, but the lecture series was cancelled before I learned what the other three are. David R. Ingham (talk) 04:45, 10 June 2012 (UTC)

As far as I know, the space is always Hilbert. "Bound states of negative energy and free states of positive energy" are eigenvectors corresponding to points of discrete spectrum and continuous spectrum, respectively. However, the former are well-defined, the latter are not (unless you enlarge the Hilbert space for some technical convenience). But anyway, eigenvectors for continuous spectrum are a science fiction; such quantum states cannot be prepared; rather, approximations to these can be prepared (just like "the delta-function" is not really a function, but its approximations are). Boris Tsirelson (talk) 05:58, 10 June 2012 (UTC)

## Image problem

I uploaded the image File:Completeness in Hilbert space.png a few minutes ago. When I realized that the yellow arrow is nearly invisible in the thumbnail rendering, I edited the image to thicken the lines. The thumbnail doesn't seem to have updated, though. Does anyone know how to force the server to generate a new thumbnail? "Purging the cache" sounds promising, but a naive attempt to do this has no apparent effect. Sławomir Biały (talk) 15:52, 11 September 2009 (UTC)

## Pre-Hilbert space

[Content transferred from Talk:Hilbert space/GA2.]

"Relative to a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a pre-Hilbert space." in the Definition section should be rewritten.

• "Sometimes is known" is, as far as I can see, factually wrong. It's an accepted mathematical definition. (Bronstein et al., Taschenbuch der Mathematik, 7. ed, 2008, p. 678)
• Also, as more than 99.9% of living human beings don't know about it, "sometimes is known" is an overstatement anyway ("sometimes is being called" would be semantically correct).  Cs32en  00:14, 14 September 2009 (UTC)
I'm afraid I don't share your objection to this particular wording. Certainly not everybody calls an inner product space a pre-Hilbert space. Indeed, most mathematicians simply call it an inner product space. Also "known as" is a very common English idiom. According to Wordnet, it is a synonym for "called". Sławomir Biały (talk) 13:17, 14 September 2009 (UTC)
Thank you for your reply! "Sometimes is known" sounds as if there would be a small community that accepts this definition, while others does not. "Known" also means, in my understanding, accepting a given statement, while "call" means to actively affirm that statement. Of couse, every pre-Hilbert space is an inner product space, but not every inner product space is a pre-Hilbert space, so the two terms are not synonymous. (The Bronstein source given above de:Taschenbuch der Mathematik is a standard work in mathematics, at least in Germany.)  Cs32en  21:17, 14 September 2009 (UTC)
I disagree with the assertion that there is any distinction between an "inner product space" and a "pre-Hilbert space". The two terms are entirely interchangeable. See, for instance, the Hewitt and Stromberg text referenced in the article. Sławomir Biały (talk) 23:32, 14 September 2009 (UTC)
Actually, your Taschenbuch also supports me in this. Sławomir Biały (talk) 23:34, 14 September 2009 (UTC)
Just looked at the Taschenbuch again. Your observation is correct, it says "is being called inner product space or Pre-Hilbert space". This would imply that any Hilbert space is also a Pre-Hilbert space, which is probably not the intended meaning of that sentence. But we would actually need another source for a clear statement that says that a Pre-Hilbert space is an inner product space that is not at the same time a Hilbert space. Or we would have to say that a Hilbert space cannot be called inner product space, which is probably unhelpful. The Taschenbuch also says (same page) that in a Pre-Hilbert space, a norm can be defined, which implies that a Hilbert space is not a Pre-Hilbert space, because in a Hilbert space, such a norm is already defined. We need further sources, but I would favor the following definitions:
• A Pre-Hilbert space is an incomplete inner product space with or without a norm.
• A Hilbert space is a complete inner product space with a norm.
But we would need more sources to clarify this.  Cs32en  00:33, 15 September 2009 (UTC)
That a pre-Hilbert space and an inner product space refer to the same thing is utterly uncontroversial. I suggest that we drop this discussion. It is not leading in a direction that will improve the article. Sławomir Biały (talk) 01:10, 15 September 2009 (UTC)
Partial agreement, as any Hilbert space is a Pre-Hilbert space, according to various sources. (Whether that is a helpful naming convention is not for WP to decide.) Still, I think that the wording "is sometimes known" is too weak. For example, de:Prähilbertraum, fr:Espace préhilbertien, es:Espacio prehilbertiano, it:Spazio prehilbertiano, ru:Предгильбертово пространство are all lemmas in their own right, not redirects. (For Japanese, Chinese, Portuguese and and couple of wikis in other languages, the lemma is "product space" or something similar.)  Cs32en  01:42, 15 September 2009 (UTC)

## Completness and the comparing with the sum of vectors.

the sentence underneath the picture with the broken line are not correct, i think.

"Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in yellow)."

The idee behind this sentence is not equivalent with completeness, and so raises confussion, or does somebody have a proof of this?...

An maybe its a good idee to clean up the discusion page. greetings S. —Preceding unsigned comment added by 157.193.53.246 (talkcontribs)

If a particle moves along a broken path with displacements v1,v2,..., then the total distance travelled is
$\sum_{n=1}^\infty \|v_n\|.$
The net displacement is
$\sum_{n=1}^\infty v_n.\,$
A space in which convergence of the former series always implies convergence of the latter is indeed complete, so I'm afraid I must disagree with you. Actually, refer to the article text right next to the image where the details are explained more fully. Sławomir Biały (talk) 01:50, 19 February 2010 (UTC)

## Example before definition

I reverted the edit that moved the example down past the definition. This example is likely to be something that all readers will be able to grasp, and so should be before the formal definition. This is the opposite order that one is used to seeing things in mathematics textbooks (which Wikipedia is not), but I do think that it is more likely to be meaningful to a larger number of prospective readers. The model for the article was based on Group (mathematics), a mathematics featured article, which also gives a toy example to clarify the concept before the definition. Sławomir Biały (talk) 23:08, 26 May 2010 (UTC)

The fact that an article is featured by no means indicates that it is perfect. And if you want to insist on this example, I'd like to point out the shitstorm that it caused: Talk:Group_(mathematics)#Definition. But the thing is, this article is already useless to the lay reader; I don't think anyone without some basic topology will be able to understand it. Also, there's already motivation and informal introduction in the lead section, the euclidean space being one of them. By beginning with the example, is left unclear why it is important: which properties of the euclidean space make it a Hilbert space? What is particularistic?
What really pisses me off is that it isn't even well-written: it defines the dot product, say that it is a special case of the inner product, and then defines the inner product. And the reader does not even why these properties are important, because the inner product hasn't been defined yet! Would you follow these demonstrations without knowing what relation they bear to Hilbert spaces? The only order that makes sense is defining a Hilbert space, defining a inner product, and then showing the example of a space that satisfies these properties.
WP:NOTTEXTBOOK only means that we shouldn't teach the subject, just present the facts. Not that we shouldn't use their conventions. And this is a pretty good convention. By beginning with the definition, you: 1 - Know what you're talking about. 2 - Avoid repetitions an relearnings. 3 - Provides a quick point of reference, to go back to it while reading the article, or afterwards. Tercer (talk) 03:09, 28 May 2010 (UTC)
You do not seem to have a clear image of the type of knowledge that non-mathematicians reading this article will have. Hilbert spaces play an important role in quantum physics. As such, a significant portion of readers looking up this article will consist of students and engineers that are getting interested in quantum mechanics. Readers with such a background will have at least some exposure to basic calculus and linear algebra, in particular, they are likely to be familiar with euclidean space and with limits in euclidean space. They are also likely to have very little (if any) knowledge of topology or linear algebra.
Hilbert spaces are modelled to generalize euclidean space in a very particular way. The current article is set up as follows. It first identifies the properties of Euclidean space, which will be kept (is linear, has inner product and is complete). (I agree that it could do a better job in relaying why these properties are important, although this will be clear to people with experience with working with Euclidean space. The second step is to give a definition that formalizes these properties. The third step is to give an example of something else that satisfies this definition.
This is a very sensible approach, since it actually provides a motivation for the definition of a Hilbert space and there by gives a much better picture of what a Hilbert space is. Mathematics texts usual don't care about a motivation for a definition, because a definition is interesting of itself, and is ultimately motivated by the richness of properties of the objects that it define. This a posteriori, type of motivation, does not make sense to anybody that is not a mathematician.
You seem to be pretty oblivious to the fact that to a large portion of potential readers abstract definitions simply won't mean anything. TimothyRias (talk) 09:21, 28 May 2010 (UTC)
You make a lot of assumptions about the readers. After all, they know linear algebra or they don't? They look for the article because they are learning quantum mechanics; but the article defines the inner product to be linear in the first argument, rather than the second (which is the standard convention in quantum mechanics). You assume that the readers will balk at the definition of Hilbert spaces, but assumes that they will follow through a lengthy digression about the euclidean space without knowing why.
You seem to ignore that the lead section already talks about euclidean space, and gives a lot of justifications about why the following definition will be interesting.. I'm not against motivations [i]per se[/i]; look at the Rigged Hilbert space article. It begins with two sections of motivation before giving the actual definition. But the motivations are actually motivations, not the oxymoronic "motivating example" now present in this article. They do not exemplify the definitions that haven't been given yet!
Even the group article does a better job in handling this. It begins with a concrete example, but it names each and every property and explain how they map to the actual definition. Tercer (talk) 16:51, 29 May 2010 (UTC)
Tercer, you are making some valid points, but you are overly critical. No one questions that there are different modes of presentation that suit different categories of readers. What you fail to realize, or at least to acknowledge, is the amount of effort that goes into creating a well-rounded article that follows its chosen philosophy (regardless of whether you or I agree with it). Simply switching the sections around isn't a good solution. The current article has undergone a number of revisions as a result of feedback from the people trying to understand it, and overwhelmingly, the bias was in the opposite direction to what you are advocating (you can find the arguments on this page and in the archives). The consistency questions and the painstaking compromises that have been reached as a result of protracted and sometimes bitter discussion are behind many long-time editors aversion to "refactoring" edits. Arcfrk (talk) 22:33, 30 May 2010 (UTC)
Arcfrk, you seem to mistake this discussion with the one you took part in 2007. I'm not advocating for a grad-level reference article (even though I'd like it, I know that's not the objective of Wikipedia), I'm only saying that the current presentation, with a highly detailed example before the definition, is awful. The article that resulted from your controversy was much more of my liking, with a separate section for the motivation written without any equations, followed by a separate section containing the actual definition. The current form of the article results from a unilateral edit by Sławomir Biały in 2009. Although I do respect his hard work, I don't think we should refrain from improving an article due to it. Tercer (talk) 23:40, 1 June 2010 (UTC)
Describing an edit made in response to comments made at the detailed good article review as "unilateral" is deeply misleading. Sławomir Biały (talk) 02:36, 2 June 2010 (UTC)
The specific point being discussed here wasn't raised. I only found a discussion about putting the definition before the history section, and merging the examples in the definition section. So this particular choice was indeed unilateral. --Tercer (talk) 04:04, 2 June 2010 (UTC)
First, I'm fairly certain that Jacob suggested this structure somewhere, if not the GAR then elsewhere, and even pointed me to the Group (mathematics) article for guidance. And even if not, the edit was part of a larger, collaborative effort that was ongoing at the time to improve the article, hence not "unilateral", regardless of whatever specific points were raised at that time. Sławomir Biały (talk) 10:41, 2 June 2010 (UTC)
I did suggest a reorganization back in the GAC ("I would definitely put this [the definition] as the first section. Accordingly, I would also suggest merging the introductory example into this section."). Of course, this was only a suggestion, and there are probably several good ways to present the material. Also, even if the group article would be perfect, this would not imply that this article here has to have the same structure etc. However, I think we had and have a consensus that the current structure is an OK one, so I suggest to leave it the way it is. Jakob.scholbach (talk) 12:01, 2 June 2010 (UTC)
I don't think WP:NOTTEXTBOOK was meant to imply that you can't give a motivation for a definition. Definitions in mathematics tend to highly abstract and difficult to follow; having a familiar concept in mind can make it much easier to understand what the definition is trying to capture. WP is not supposed to teach the subject but an article should at least be understandable to the people who are likely to read it. In this case I think it's reasonable to assume that a reader will have some understanding of calculus and linear algebra, but in order to reach as broad an audience as possible nothing more should be assumed. Given that, giving a motivating example before the definition is entirely reasonable and desirable. I only wish more math articles took this approach.--RDBury (talk) 18:15, 2 June 2010 (UTC)

## Self-duality

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

See section "Duality", where the matter is discussed at length. Sławomir Biały (talk) 10:29, 9 June 2010 (UTC)

## Inclusion of Folland reference

Sławomir Biały: I'm happy you included Folland's book as a reference. This work is a favorite of mine for a clear exposition of the subject. Brews ohare (talk) 17:59, 24 June 2010 (UTC)

## Euclidean space

In a pair of edits (here and again here), the term Euclidean space was removed in favor of other possibilities: "finite dimensional Euclidean space" and "n-space". However, I think that Euclidean space should be restored. For a totally non-mathematical audience, this conveys precisely that Hilbert spaces are spaces in which one can still perform Euclidean geometry, whereas "n-space" conveys nothing meaningful whatsoever to such an audience, and "finite dimensional Euclidean space" seems overly complicated. I realize that there are mathematicians that also consider infinite-dimensional Euclidean spaces (in fact, these are what we call "Hilbert spaces" here), but the finite dimensional case is still the primary use of the term (both within mathematics, and certainly outside it). Anyway, we needn't be overly concerned for those who "know better": they will certainly be able to cope as well. I think that perhaps these edits have missed the point that the lead section of an article should be a general-audience description and not one that is intended to be a mathematically precise characterization. Sławomir Biały (talk) 14:26, 5 October 2010 (UTC)

This seems like an excellent idea to me. (Thenub314 quickly checks that he wasn't responsible for originally adding the extra verbiage... ok, good I wasn't. :) ). I am going to be bold and go ahead and make this change. Thenub314 (talk) 17:29, 5 October 2010 (UTC)
Looks like I was beaten too it. Thenub314 (talk) 17:33, 5 October 2010 (UTC)
That was I, in response to various problems pointed to by WT:MATH. — Arthur Rubin (talk) 20:26, 5 October 2010 (UTC)

## Good work here

I'm impressed that I clicked on the wikilink of a maths term I didn't understand, read the first paragraph of this article, and now understand enough to go back and finish the article I was reading. This is how wikipedia should be! Thanks guys. --Physics is all gnomes (talk) 00:09, 28 December 2010 (UTC)

I would just like to say thanks to this talk page I finally get Hilbert Spaces, much easier to glean concepts from a few arguments than the article itself. I'm a PhD Student Cheers :) --78.86.197.227 (talk) 17:06, 4 May 2011 (UTC)

## Notes-Section

In the "Notes-Section" at point 46 is referred to: Dunford & Schwartz 1958, II.4.29 Well, I looked for that reference, but I cannot find it. Here is the book at Amazon: [1] There is simply no section "4", but there are many... Have I picked the wrong book or is the reference incorrect? --Vilietha (talk) 14:44, 25 April 2011 (UTC)

The II refers to the chapter number. Volume 1 has chapters I-VIII, and Volume 2 has IX-XIV. Sławomir Biały (talk) 15:03, 25 April 2011 (UTC)

## Angle brackets

There are three kinds of such brackets in Unicode, see Bracket (mathematics). The correct characters for mathematical angle brackets are ⟨mathematical left/right angle brackets⟩ (U+27e8 and U+27e9). The HTML entities lang and rang resolve to left/right-pointing angle bracket (U+2329 and U+232a), which are deprecated by Unicode because they are canonical equivalent to Chinese punctuation (U+3008 and U+3009). For a proof of why we must not use lang and rang entities, see the revision 2011-08-22T05:57:55 by Headbomb. I doubt he replaced them intentionally, so I conclude that some Unicode aware software may replace them automatically with the characters inappropriate for mathematical text. The root of the problem is that lang and rang don't have the correct semantic information.

Hereby, in order to avoid any further debate, I replace all angle brackets with LaTeX as required by convention. bungalo (talk) 08:56, 30 August 2011 (UTC)

Previous discussion: Talk:Hilbert_space/Archive_2#Angle_brackets --LutzL (talk) 11:00, 30 August 2011 (UTC)

Thanks, I had forgotten this earlier discussion. It seems I was wrong in assuming that &lang; and &rang; would resolve to the correct character just because they are named html entities. I've gone back to Ybunalobill's version before the introduction of LaTeX. Sławomir Biały (talk) 11:16, 30 August 2011 (UTC)
Then one should also mention explicitly that, in math articles in en-wiki, formulas inside text should preferably realized as html formulas. With unicode characters and not html entities, since html entities are always replaced by some bot, and in this case with the wrong characters.--LutzL (talk) 12:52, 30 August 2011 (UTC)
In the light of the said above, can someone explain to me why User:Slawekb reverted my changes when I replaced with the mathematical Unicode characters? It seems that both of you agree that what is currently used is incorrect (namely, the HTML entities which resolve to the incorrect character). Leaving it as-is is waiting for trouble. The correct way is to use the mathematical left/right angle brackets. In case someone doesn't see the math symbols, I would say that I expect of someone who reads math articles to have math fonts installed. There are some good freely available. I don't understand why others who do have the correct system configuration should see Chinese. bungalo (talk) 07:33, 31 August 2011 (UTC)
Sorry, I made a mistake. I had intended to restore your version, not mine. Sławomir Biały (talk) 11:51, 31 August 2011 (UTC)

## Contrast with vector space

Can we please avoid incorrectly suggesting that every hilbert space is a vector space? Vector spaces are generated by finite combinations of bases, while the finite restriction is lifted in hilbert spaces. I know it is a technical detail, and colloquial verbal usage frequently ignores the difference -- but let's be correct. --Liuyipei (talk) 08:43, 29 February 2012 (UTC)

A Hilbert space is a special instance of a vector space, with additional structure. Given a Hilbert space, one can speak of its vector space (or Hamel) dimension or its Hilbert space dimension. For instance, a Hilbert space with a countable Hilbert basis, i.e. separable Hilbert space, need not have countable vector space dimension. Mct mht (talk) 09:06, 29 February 2012 (UTC)
Liuyipei,
You should review your definition of a vector space, which is a module over a field.
As a counter-example to your claim, consider the vector space of the real numbers over the field of rational numbers. Are you really suggesting that the square-root of two is a finite rational linear combination of rational numbers?
09:27, 29 February 2012 (UTC)
Eh. Seems to me the appropriate question is "is the reals, as a module over the rationals, finitely generated?" Mct mht (talk) 10:29, 29 February 2012 (UTC)
Hilbert spaces are vector spaces. See any book. Sławomir Biały (talk) 11:41, 29 February 2012 (UTC)
Briefly, in the definition of a vector space, finite linear-combinations appear in the algebraic-closure axioms (linear combinations of elements in a vector space remain in the vector space). There is no negation of the existence of infinite linear-combinations. 13:11, 29 February 2012 (UTC)

## Trivia/popular culture/architecture?

At Goettingen, the math building's foyer, where students normally first enter, is called "Hilbertraum" [Hilbert room/space]. According to the German Wik, other German universities also have Hilbertraeume. should this be incorporated into our article? Kdammers (talk) 03:38, 29 April 2012 (UTC)

That seems like a fairly miscellaneous fact that doesn't really belong in the article. Sławomir Biały (talk) 10:04, 29 April 2012 (UTC)
Especially since it is quite possibly more a reference to the Hilbert hotel paradox, than Hilbert space. Without a reliable source to tell us which it would not be addable anyhow.TR 13:45, 29 April 2012 (UTC)
That's a little unlikely, as "Hilbertraum" is the German word for Hilbert space (in the sense that we mean here). No German speaking mathematician would associate this with Hilbert's grand hotel. Sławomir Biały (talk) 14:05, 29 April 2012 (UTC)
Right. All of us, once we had gotten that far, knew exactly what the "joke" was, though I don't recall it ever being mentioned in class. Kdammers (talk) 13:03, 3 May 2012 (UTC)

## Is Hilbert dimension a common term?

Hilbert dimension is an algebraic concept, from ring theory. Hilbert space dimension seems to be more common for the dimension of a Hilbert space in my opinion. --Chricho ∀ (talk) 16:05, 3 August 2012 (UTC)

Hilbert dimension also commonly refers to the dimension of a Hilbert space. However, because of the ambiguity of the term, the article Hilbert dimension should be a disbiguation page rather than a redirect here. Sławomir Biały (talk) 16:49, 3 August 2012 (UTC)
Done: Hilbert dimension. --Chricho ∀ (talk) 13:42, 8 August 2012 (UTC)

## why replace the correct formulation?

 “ "(...) The Occam's razor is a principle of parsimony, economy, or succinctness used in logic and problem-solving. It states that among competing hypotheses, the hypothesis with the fewest assumptions should be selected. (...)" Differences between last page I reverted and current version ” —"Hilbert space" article

Hallo there Prof. B. Tsirelson,

I have replaced "(...) the possible states (more precisely, the pure states) of a quantum mechanical system (...)" with "(...) the quantum states of a system (...)" in an attempt to make text more readable.
1. If "possible states" can be "more precisely" written with "pure states"... then why not write directly "pure states"?
2. And if "pure states" redirects to the "quantum state" article (#redirectquantum state)... why not write directly "quantum state" then?

I hope my explanation helps.
Please let's try to have a nice and relaxed weekend.
Cheers.
M aurice   Carbonaro  09:53, 19 April 2013 (UTC)

Quantum states can be pure or mixed, and only a pure state can be described by a single vector in a Hilbert space. Mct mht (talk) 10:24, 19 April 2013 (UTC)
Yes, this was my point: only a pure state can...
And yes, I also wish to make text more readable; but I do not think that shorter is always more readable. Physically, most vectors have slim chance to really appear among the states $\psi(t)$ for some instant t; they are possible states, and maybe the word "possible" helps to some readers to understand the point correctly. By the way, I like the idea of falsifiability, but I did not take the hint: how does it applies here? And anyway, best wishes to you too. Boris Tsirelson (talk) 11:07, 19 April 2013 (UTC)
Hallo there Mct mht (talk) and Professor B. Tsirelson (talk),
thanks to both of you for answering thoroughly.
I apologize for the edit: I just realized that in
point # 2 (And if "pure states" redirects to the "quantum state" article (#redirectquantum state)... why not write directly "quantum state" then?)

Regarding the falsifiability option: I gave a hint about it in order to set a demarcation criterion: there are some aspects in physics that sound very odd. And it seems like I am not the only one to be mesmerized about them. I have been late in answering: please bear with me (this means you should NOT get undressed...) Cheers.   M aurice   Carbonaro 06:57, 23 April 2013 (UTC)

## Speaking about bases in the introduction

Talking about the representation of elements within a Hilbert space, the introduction says:

An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane.

In fact this would also hold for any orthogonal basis, and even for any vector set that forms a basis, without the need for them to be either orthogonal nor normal. Besides this, I feel the analogy with the Cartesian plane is a little bit low-level, and I think it would be more appropriate in the article explaining what a base is.

I would either remove the entire sentence, for it applies to any vector space and this is not a particular property of Hilbert spaces alone, or would at least remove the word "orthonormal".

Using the same reasoning, I would like to reconsider this affirmation:

Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.

If the chosen base is not orthogonal, could we say that linear operators stretch the space in mutually perpendicular directions?

Elferdo (talk) 15:35, 22 July 2013 (UTC)

Given that it is already a good article, please be careful. Your taste may differ from that of others. You have already made a change that I find doubtful. And your proposal above is doubtful: does it make the intro more accessible? Technical details may be not so good for the intro. Boris Tsirelson (talk) 15:58, 22 July 2013 (UTC)
Except that Cartesian coordinates are in an orthonormal basis, not just an orthogonal one, so I find your first objection questionable. Your second objection seems to be missing the point. Sławomir Biały (talk) 07:34, 23 July 2013 (UTC)
Ok, I'm sorry, I apologize for the modification. I will be more careful in the future.
My first objection was not that Cartesian coordinates are not orthonormal, I did not say that. My argument is that the sentence that I quoted could be somewhat misleading because that is not the only way to represent vectors (which does not exclude the fact that the affirmation is true). If, however, accessibility is preferred over completeness, as Tsirel implies, then I'm ok with it.
I don't understand why Sławomir suggests that my second suggestion misses the point. Again, I argue that orthonormality is just one of many possible choices of base. So my question is if the sentence is as general as can be, and if it would apply to other choices of base. Again, I agree that the introduction is accessible and clear, but I fear that it is not as complete and accurate as could be.
In conclusion, I do not object to the correctness of the introduction, but I do object to its completeness. I'm not an expert in the field, however, so if you both feel that I'm wrong, or that this is a good trade-off for the sake of understandability, then I'm ok with that.
Elferdo (talk) 14:06, 23 July 2013 (UTC)
What choice of basis? Sławomir Biały (talk) 14:50, 23 July 2013 (UTC)
Well, whatever basis happens to be the one over which vectors are represented, which need not be the canonical one, neither an orthonormal one. Assuming orthonormality is good for an introduction to the subject, but it is, in my opinion, too much to assume for the sake of completeness. Again, I will gladly accept this assumption for the sake of clarity, but I argue that the introduction is not as general and complete as it could be. Elferdo (talk) 16:41, 23 July 2013 (UTC)
I don't understand why you seem to think that spectral theory has something to do with a choice of basis. It does not. Sławomir Biały (talk) 00:51, 24 July 2013 (UTC)
I feel this is getting a bit off-topic, so I will be happy to close this discussion. One last note, however, back to my first comment:
Talking about the representation of elements within a Hilbert space, the introduction says:
An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane.
I don't believe this is a statement about spectral theory at all, and I don't understand why you bring spectral theory into the discussion. In my opinion, "specifying an element of a Hilbert space with respect to a set of coordinates" has everything to do with a choice of basis. Elferdo (talk) 11:08, 24 July 2013 (UTC)
I was objecting you your second objection in the original post: the one starting "Using the same reasoning, I would like to reconsider this affirmation..." The affirmation in question has everything to do with spectral theory, and nothing to do with bases. Sławomir Biały (talk) 13:23, 24 July 2013 (UTC)
Ok, sorry. Like I said elsewhere I'm not an expert in Hilbert spaces, so it was rather a doubt than an objection. I realize now that the intention is to give specific examples, "good cases", rather than trying to be rigorous or complete. It is not the way I would expect the subject to be introduced, but I agree that it is a matter of taste, so we can declare the issue resolved here.
This notwithstanding, I disagree with you with respect to the [non]-relationship between spectra and space bases (let's say it's a weak disagreement, since I acknowledge that you may be more expert than me in this field). As I understand it, the spectrum of a linear operator is nothing but the set of eigenvalues of its matrix representation. This, indeed, is dependent upon the chosen basis of the space that the operator transforms.
Quoting http://en.wikipedia.org/wiki/Eigenvector#Eigenspace_and_spectrum:
There is no unique way to choose a basis for an eigenspace of an abstract linear operator T based only on T itself, without some additional data such as a choice of coordinate basis for [the space on which T operates].
Also from http://en.wikipedia.org/wiki/Change_of_basis#Change_of_basis, I understand that a single operator has different matrix representations upon different bases of the space. I cannot conclude taxatively that the eigenvalues must then be different for different bases, but my intuition makes me think so, please correct me if I'm wrong. Elferdo (talk) 14:24, 24 July 2013 (UTC)
It's well-known that the spectrum of an operator is independent of the choice of basis, as are its eigenspaces. Matrices do not enter the picture at all. Sławomir Biały (talk)
I think this is already a discussion of the matter itself, rather than a discussion about the article, so I suggest that we either move this discussion to our user talk pages or close it. Also, I would appreciate it if you would support your arguments with references, rather than relying on "well-knownness". I'm curious to know, however, how you calculate eigenvectors and eigenvalues if "matrices do not enter the picture at all".Elferdo (talk) 14:49, 24 July 2013 (UTC)
Please consult any textbook on linear algebra. Sławomir Biały (talk) 15:17, 24 July 2013 (UTC)
I think that wasn't really helpful from you. Let's close the discussion here.Elferdo (talk) 16:17, 24 July 2013 (UTC)

BTW, the entire passage in the lead about linear operators looks very dubious to me. Over complex numbers a linear operator is not necessarily self-adjoint, whereas over real numbers it may have an empty spectrum, but even if sufficient number of eigenvectors exist to form a basis, they are not necessarily orthogonal. I wonder how this crap managed to find its way into this so named “good” article in spite of all this crowd of experts here. Incnis Mrsi (talk) 14:47, 23 July 2013 (UTC)

The statement is specifically about self-adjoint operators (hence "in good cases"). We're trying to convey a taste of the subject in the lead without going into such technical details. Sławomir Biały (talk) 14:50, 23 July 2013 (UTC)
I agree with Slawomir. First, Intro should be Intro. Second, the whole article is basically an intro into a large domain, a kind of survey. Details are scattered in more narrow articles. Orthonormal bases (in Hilbert spaces) are more notable than arbitrary bases; likewise, Hermitean operators are more notable than arbitrary operators. Boris Tsirelson (talk) 05:43, 24 July 2013 (UTC)

## no cloning theorem

This article never mentions the no-cloning theorem or its cousins.It seems to me that this is an important property that *all* (finite-dimensional) Hilbert stpaces have. Viz, you cannot clone some abitrary vector in a Hilbert space, you can only clone the basis vectors (which in turn implies that basis vector must be orthogonal) ... the no-deleting theorem implies that the basis vectors form a complete set ... I mention this because 1) in dagger compact category, these two theorems are used to define the basis, and 2) the dagger compact cat is complete in Hilbert spaces: any theorem that holds for hilbert spaces holds for any dagger compact category, in general ... there should be some blurble of this in this article, but I'm not feeling "bold", as they say. (this article already being quite extensive and thorough -- good job!)

Hmm. I guess a point of confusion would be "what does it mean to clone", as a college student would say "easy, whip out pen and paper, and write it down twice". So somehow, need to get across the idea that if one has two identical hilbert spaces, and has a vector in one, there is no way to take a general vector in one and copy it into the other (short of specifying an infinite number of decimal places in some basis ... Hmmm ... I don't see a good, simple, pedestrian explanation at the moment...) User:Linas (talk) 19:05, 27 November 2013 (UTC)

This all seems very off topic in a general article on Hilbert spaces. Sławomir Biały (talk) 19:55, 27 November 2013 (UTC)
As far as I know, the very first (now rather naive, probably) form of the no-cloning theorem was (mathematically) just the fact that the map $H \ni x \mapsto x \otimes x \in H \otimes H$ is not unitary. This is about Hilbert spaces, but mathematically too trivial for being mentioned here. And the dagger compact category (and all that) is too far from the general theory of Hilbert spaces. Boris Tsirelson (talk) 21:08, 27 November 2013 (UTC)