Talk:Hilbert space/Archive 1

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experts do not use Knowledge. 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! -

1949:-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. 1714:

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.

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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". (

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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

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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.

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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.

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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. Knowledge 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.

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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.--

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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

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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.

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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.

1573:. 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? 2625:

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.

3893:"A vibrating string can be modeled as a point in a Hilbert space. The decomposition of a vibrating string into its vibrations in distinct overtones is given by the projection of the point onto the coordinate axes in the space." Having looked at the article "Overtone" it seems to me that the correct term as used here should be harmonic, not overtone. I could be wrong. Please comment. 3748:, 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!) 802:"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. -- 271:

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. -

2199:. 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 3865:(although most of these were complaints about the article quality) so I figure a connection could be made here. Whatever ... I often get over-excited by ideas ... looking over this article, I see its very long; there is a section on QM, but indeed, there seems to not be much point in taking up more space there. Although perhaps the sentences about 2560:"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"? 2583:"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. 113:. 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. 4249:)" 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. 3861:

Such is life. Yet here we are-- it has spawned an entire industry, not only of mutiple academic journals but also venture-capital-funded companies. Its all anchored on this "property" of Hilbert spaces, and figured it deserved a 1 or 2 sentence mention. There is a burning interest in the topic; I just archived 40 comments at

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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.

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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.

567:|| 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). - 5301:

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

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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

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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

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The direct sum of Hilbert spaces as defined in the article doesn't need a restriction on the index set. By definition, it's a linear subspace of the direct sum of the H_i (in the category of vector spaces). It is the linear subspace consisting of those elements of the direct sum space such that the

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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,

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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

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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

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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

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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

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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

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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

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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

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Similar confusion can be found in the definition of unbounded operators. There is no reason to formally define an unbounded operator. There are operators (linear ones). They may be continuous (in which case we also call them bounded). It is not a particularly interesting situation when operators are

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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

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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

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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

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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,

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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

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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

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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

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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

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Further to this, I'm a physics undergraduate - third year, near the top of my class at a good university, and I find this article and many others indecipherable. It strikes me very strongly that the only people these things are written for are people who are 'supposed' to already know all about it -

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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

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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.

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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

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In the same section, I don't understand the sentence "only countably many terms in this sum 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

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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

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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

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is finite. This expression makes sense as an extended real number even if I is uncountable, since it is a sum of nonnegative terms. In fact it can only be finite if all but countably many of the x_i are zero. This expression also defines the (square of) the norm, and the inner product is defined

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When someone slipped me a copy of the teleportation QM paper when it first came out, it was fore-head-slappingly obvious: any undergraduate could have gotten it; I was embarassed for the authors, it was too simple. I sort-of kicked myself for not having spotted it myself, back when I was school.

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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

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I did suggest a reorganization back in the GAC ("I would definitely put this 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,

1626:

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,

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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

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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

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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

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The other conventions are mentioned in the footnote. I think that is about the right degree of prominence. If someone is worried about an exam, they had better use the conventions that their professor and/or textbook adopt, not what some Knowledge article tells them to do (while mentioning those

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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

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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

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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

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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

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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

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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

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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

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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

710:

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

346:

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

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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

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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

101:

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

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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,

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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

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Even though the aim is those who come to learn and not those who already know what they are reading, and even though it is a big matter of consideration if the article can mislead an important portion of readers (whichever their nature), please excuse my commentary on exams, which meant not to be

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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

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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

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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

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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

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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

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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

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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

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This notwithstanding, I disagree with you with respect to the -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

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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

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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

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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

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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 Knowledge), I'm only saying that the current presentation, with a highly detailed example before the

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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

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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

400:

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

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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

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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

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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

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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).

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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,

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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

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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

890:

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

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Hi, Sławomir, thank you for your interest. We are talking about the most widely used convention in Hilbert spaces, which is not the general mathematical convention, that surely must appear in mathematical books alone or together with a succinct footnote of the kind. However, Knowledge is an

3545:

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

2286:

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

3739:

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

2616:

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

5144:

deserve something beefier, but we have that already; the article on it. A sentence, including the visual appearance, togerther with a link to the article should suffice. Other notations we probably don't have articles on, so those deserve some space. It is also not uncommon to use the

4479:

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

3253:

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.

1617:

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.

687:), then the first function is no good: it's not square integrable. If your domain is something smaller, 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). - 4630:

Note about linearity conventions is clearly insufficient and causes confusion among students from major areas of knowledge, like physics. The call has to be more informative and visible, with especial care on the antilinear argument, which breaks the symmetry... and many exams too.

3426:

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.

123:

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.

5392:=equation for (a different mechanism). I'm am pretty sure from (other reading) that the two notations have different in meaning (they are from different areas of mathematics, apply to functions of different classes, and shouldn't be mixed without citing which is being used). 4119:

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,...:

2263:

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

3482:, 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. 2244:

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.

354:

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

3299:

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

503:

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

4453:

Some early feedback, some acronyms are introduced out of the blue (PDE for partial differential equation, L 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

5059:

I hope that everything is clearer now so that we don't need to extend this discussion. I am looking forward to reach consensus on the proper paragraph about the topic and close the thing in the least possible time. I will suggest a little text in the upcoming days.

4480:

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

1677:

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

950:

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.

739:, 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. 523:

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

2639:

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.

2438:

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.

4982: 2563:"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)." 5258:, 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 3441:

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.

3231:

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

1509:

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.

158:? 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. 5352:

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.

5086:. Having those present will distract from the main point, namely that a Hilbert space is a complete normed space, where the norm is induced by a sesquilinear inner product. It might be appropriate to have a separate section on notation, introducing 1868:

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.

1579:

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.

2287:

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

4336:

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 =(2π)δ(p-p') =:

2172:

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?

1303: 515:

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

2566:"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. 2070:

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.

452:"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." 1285:

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,

4774: 1117:

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.

4109:

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

1799:

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.

883:

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

4565:

I would like to add that there is no reason not to discuss both in the article. However, since they are not the same thing, that would actually involve writing new text, rather than replacing "Bergman" by "Bargmann" in the old text.

1838:

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.

3961:

with no limitation on the index set. This is probably fine, but I suspect that the notion of nets and net convergence is needed to cover the uncountable case. Some quirks appear "already" at the countably infinite level, see Conway -

3704:

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

3673:, 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. 2466:

At Goettingen, the math building's foyer, where students normally first enter, is called "Hilbertraum" . According to the German Wik, other German universities also have Hilbertraeume. should this be incorporated into our article?

969:

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

4368:. 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 | 5183:"The adjoint of a densely defined unbounded operator is defined in essentially the same manner..." WTF does "essentially the same" mean? Either it is the same - then state so. Or it is not the same, then describe the difference. 2372:

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

1189:

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

937:

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.

295:

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.

5046:

encyclopedia (as it is obvious, and as it is stated as one of its Five Pillars). Articles have to properly cover all major realities; let me introduce you to this reality and its importance, which needs proper coverage.

1659:

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.

1672:

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 per se; look at the

2569:"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. 231:

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.

106:). 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. 2104:"...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 1244:

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.

1197:

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:

798:

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.

722:

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. -

4264:

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. (

261:

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."

175:

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.

5134:

At any rate, I introduced a footnote of the "visible type". (This type of footnote can be used for other present regular footnotes.) Maybe this is adequate. I'll also make a strategic move of the footnote.

3837:

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.

3752:

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...)

4229:

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". (

435:

that is, professional physicists. There is far, far too much jargon that I suspect even many physicists wouldn't be familiar with. I strongly suspect that this is the realm of postgraduate mathematics.

265:

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.

5395:

Saying some contrivance ties all the topics mentioned by a grand unified theorem: simply isn't true. I would like to dispel contrivances by arguing but instead rest with the above request for change.

4039:

in the usual manner. The norm induces a metric, and t it's easy to prove that metric is complete. Nowhere do nets appear (and sequences only appear in verifying completeness of the metric space).

3835: 2518:

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.

222:

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.

96:

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.

5402: 1444: 3397:

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.

3268:

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.

3216:

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.

3224:

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".

1503: 1049:

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

4036: 3473:

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 .

1627:

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.

4828: 3688:

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.

2085:

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 :) --

500:"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." 4852: 3956: 83:. 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. 2985:

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

1691:

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.

1342:"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)." 1306:

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.)

644: 4359:

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 |

926:

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. –

363:

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

4214:, 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. 4202:

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: "

5187:

not continuous and not having a particular property is most of the time useless. It would make sense, however, to generalize from fully defined to densely defined operators.

4183:

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. (

4166:

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,...: -->

2030:

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:

1569:

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:

5321:

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.

4126:

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?

5344:

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

2667:

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.

3012: 5214:

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:

5131:(cited) for one choice or the other may ignite a fire. But, perhaps the audience (and set of editors) of this article is less crackpot-infested than that of some other. 4662: 4374:

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.

4321:

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).

1532:

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

4588:

It looks as if one of the two parts of this image is upside-down. The summation of the bottom part of the diagram should surely give the inverse of the top part ?

4167:

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.

4458:

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!).

887:

The applications of Hilbert spaces (quantum mechanics, PDEs, stochastic processes) should be in the “Introduction” section rather than the “Definition” section.

146:

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

208:

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.

325:

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). -

150:, 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 1026:

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

3496:

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.

2031: 2805: 390:

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. -

2208: 762: 464: 849:

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

5149:

notation for the inner product and the action of a linear functional (with the order of arguments allowed to be flipped as suits the occasion).

3665:

BTW, the entire passage in the lead about linear operators looks very dubious to me. Over complex numbers a linear operator is not necessarily

984: 5274:...with further technical details coming in gradually lower down. The whole article has similar problems, however, which is why I tagged it. 5127:

My experience is that too much discussion on such matters quickly deteriorates to a discussion about which convention is "right". Even giving

3787:

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

3546:"well-knownness". I'm curious to know, however, how you calculate eigenvectors and eigenvalues if "matrices do not enter the picture at all". 1544:), 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 4245:

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

459:

gives a general example of unbounded operator on any infinite dimensional normed space into any non-zero normed space, so this is not true.

5194: 2659: 1364: 2534:

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.

1636:

You seem to be pretty oblivious to the fact that to a large portion of potential readers abstract definitions simply won't mean anything.

3024:

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.

812:

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. -

4595: 4515: 3333:

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.

2086: 774: 508:

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

3869:

could be cut ... if for no other reason than the POVM article is horrid. Happy Thanksgiving! And again, excellent job on this article!

3363:

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:

2729:

Hilbert dimension also commonly refers to the dimension of a Hilbert space. However, because of the ambiguity of the term, the article

1345:

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

2160: 5406: 5061: 4990: 3463:

eigenvalues of its matrix representation. This, indeed, is dependent upon the chosen basis of the space that the operator transforms.

1082:

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

249:

the article was wrong. The implication was that if you are a crackpot, of course your articles about Hilbert spaces will be wrong. -

1834:

First, I'm fairly certain that Jacob suggested this structure somewhere, if not the GAR then elsewhere, and even pointed me to the

2144:

suggestion: "Some older references define Hilbert Spaces to be strictly infinite dimensional. e.g. Kolmogorov and Fomin (1970)."

3236:

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

118:

In this space, a point represents the entire physical system." I would suggest not relying too heavily on Sarfatti's writings.

2076: 670:

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?

4464: 4104:

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.

879:

standards and style. In my opinion, this is a Good Article, however, I would like to make a few suggestions for improvement:

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I don't understand why you seem to think that spectral theory has something to do with a choice of basis. It does not.

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There is simply no section "4", but there are many... Have I picked the wrong book or is the reference incorrect? --

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not say that relativity is wrong. I use relativity. Ditto with quantum theory. So what is Lethe talking about? BTW:

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Quantum states can be pure or mixed, and only a pure state can be described by a single vector in a Hilbert space.

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The section “Operations on Hilbert spaces” has only one sentence. Can this be put into another section or expanded?

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It states that among competing hypotheses, the hypothesis with the fewest assumptions should be selected. (...)"

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Eh. Seems to me the appropriate question is "is the reals, as a module over the rationals, finitely generated?"

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quickly checks that he wasn't responsible for originally adding the extra verbiage... ok, good I wasn't. :)

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Off-topic, maybe, but .. coming from a physics background, I eventually learned that not all operators are

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I didn't claim that you were a crackpot because the article was wrong. I claimed that you were a crackpot

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is an abstract spatial concept used by mathematicians to consider the relations between certain types of

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Yes, you are, of course, correct. I read the section (and also Conway) too quickly and missed the point.

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from physics, and possibly other familiar notations from quantum mechanics like amplitudes and traces.

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introduction is accessible and clear, but I fear that it is not as complete and accurate as could be.

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It is not a exceptional case; much mathematics is developed with that other (hidden) convention.

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As noted in the mathematics Manual of Style, the use of the first person (“we”) should be avoided.

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states, and maybe the word "possible" helps to some readers to understand the point correctly.

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Another option: "... is complete, meaning that it is not a dense subspace of a larger space."

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I would suggest giving a list of basic references, under a “Further reading” appendix section.

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great input - if I was in a graduate class in topology this article works, but MOST are not

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I don't think the "Definition" section is an appropriate place to discuss other conventions

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

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The II refers to the chapter number. Volume 1 has chapters I-VIII, and Volume 2 has IX-XIV.

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

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

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Hereby, in order to avoid any further debate, I replace all angle brackets with LaTeX as

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is a principle of parsimony, economy, or succinctness used in logic and problem-solving.

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That seems like a fairly miscellaneous fact that doesn't really belong in the article.

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described via "ancillas", "effects", "quantum operations", "quantum instruments" etc).

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as a reference. This work is a favorite of mine for a clear exposition of the subject.

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Well spotted. I have corrected it. I hope the original artist doesn't mind. — Cheers,

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Ok, I'm sorry, I apologize for the modification. I will be more careful in the future.

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The lack of precision in many mathematical texts of Knowledge really is astonishing.

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accessibility is preferred over completeness, as Tsirel implies, then I'm ok with it.

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just like the 95% of the world who believes a picture can tell more then 1000 words.

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implies that the basis vectors form a complete set ... I mention this because 1) in

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but I agree that it is a matter of taste, so we can declare the issue resolved here.

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You should review your definition of a vector space, which is a module over a field.

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Well I added some stuff to the article which might help. Let me know if it does. -

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

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That should have read "if a closed linear operator is defined" etc. I'll fix it.--

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unfortunately doesn’t mention why (probably because it was written by physicists).

5238:. It's just too much. My feeling is that you need something much more general: 5016:

In some conventions, inner products are linear in their second arguments instead.

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You have not guessed right. The criterion for Cauchy convergence of a sequence {

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A Pre-Hilbert space is an incomplete inner product space with or without a norm.

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If you wish to start a new discussion or revive an old one, please do so on the

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Regarding the falsifiability option: I gave a hint about it in order to set a

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seems to be more common for the dimension of a Hilbert space in my opinion. --

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I think that wasn't really helpful from you. Let's close the discussion here.

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Sorry, I made a mistake. I had intended to restore your version, not mine.

2506:. Without a reliable source to tell us which it would not be addable anyhow. 580:

Very true, but how is the difference of two Hilbert space vectors defined?

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Isomorphic as what? And are separable and inseparable spaces isomorphic?

371:. Once you understand those, perhaps this article will become accessible. 756:"Since all infinite-dimensional separable Hilbert spaces are isomorphic" 1540:

order that one is used to seeing things in mathematics textbooks (which

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as a pre-Hilbert space." in the Definition section should be rewritten.

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About nonseparable spaces: are you sure? The usual representation of a

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

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An maybe its a good idee to clean up the discusion page. greetings S.

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I have placed the article “On hold” for the changes to be implemented.

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This all seems very off topic in a general article on Hilbert spaces.

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reference, to go back to it while reading the article, or afterwards.

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It follows from properties 1 and 2 that a complex inner product is

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promising, but a naive attempt to do this has no apparent effect.

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most of what's in this page would be looking it up on Knowledge.

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I've just discovered my book uses (f,g)=equation and <f,g: -->

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Yes your right. I got confused by the bad habits of physicists. (

4121:, where the pi's are the momenta and thus continuous parameters.) 1053:. Also "known as" is a very common English idiom. According to 102:

think Knowledge can do a lot better than this (and it has-- see:

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A mathematician I used to work with, Kane Yee (best known for a

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I have rewritten the first paragraph to address this concern.

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Especially since it is quite possibly more a reference to the

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In the light of the said above, can someone explain to me why

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A Hilbert space is a complete inner product space with a norm.

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than arbitrary bases; likewise, Hermitean operators are more

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Differences between last page I reverted and current version

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Describing an edit made in response to comments made at the

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http://en.wikipedia.org/Eigenvector#Eigenspace_and_spectrum

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should be a disbiguation page rather than a redirect here.

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If a particle moves along a broken path with displacements

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That was I, in response to various problems pointed to by

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The section "bases" is reproduced verbatim in the article

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is a standard work in mathematics, at least in Germany.)

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Thankfully, I managed to find a satisfactory explanation

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Actual infinity#Aristotle's Potential-Actual Distinction

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I am going to be bold and go ahead and make this change.

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http://en.wikipedia.org/Change_of_basis#Change_of_basis

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There are three kinds of such brackets in Unicode, see

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Move the examples of the space to the examples section.

3830:{\displaystyle H\ni x\mapsto x\otimes x\in H\otimes H} 3162:. I have been late in answering: please bear with me ( 1335:

Completness and the comparing with the sum of vectors.

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Maybe I'll comeback for a more detailed pass later. (

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the correct system configuration should see Chinese.

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also says (same page) that in a Pre-Hilbert space, a

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is countable. Am I misunderstanding the sentence? --

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Please let's try to have a nice and relaxed weekend.

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quantum measurement (which is much simpler that the

235: 3912:In the article, the general expression is given as 4976: 4822: 4768: 4030: 3950: 3829: 3063:I apologize for the edit: I just realized that in 3006: 2420:Hilbert spaces are vector spaces. See any book. 1497: 1438: 638: 4286:I have added a paragraph with a reference to the 3669:, whereas over real numbers it may have an empty 3160:I am not the only one to be mesmerized about them 154:should be aimed at the lay person? or how about 4834:where the case of equality holds precisely when 4645:in its first argument. For all complex numbers 3562:Please consult any textbook on linear algebra. 2982:Yes, this was my point: only a pure state can... 2843:"(...) the possible states (more precisely, the 1209:But we would need more sources to clarify this. 679:Well, I'm not sure what your domain is, if it's 457:http://en.wikipedia.org/Discontinuous_linear_map 4782:The inner product of an element with itself is 3059:thanks to both of you for answering thoroughly. 1439:{\displaystyle \sum _{n=1}^{\infty }\|v_{n}\|.} 236:Sarfatti's defense of his Hilbert space article 5152:Thinking closer, this discussion on notation 2204:don't have the correct semantic information. 1498:{\displaystyle \sum _{n=1}^{\infty }v_{n}.\,} 8: 4968: 4949: 4931: 4912: 4894: 4856: 4811: 4799: 4760: 4741: 4732: 4713: 4704: 4666: 4085:relate to complex Hilbert spaces? — Cheers, 4019: 4005: 1430: 1417: 4031:{\displaystyle \sum _{i\in I}\|x_{i}\|^{2}} 2710:is an algebraic concept, from ring theory. 2227:Talk:Hilbert_space/Archive_2#Angle_brackets 1966:This seems like an excellent idea to me. ( 1387:,..., then the total distance travelled is 711:have all acknowledged it's amazing utility. 5396: 5188: 4589: 4509: 3164:this means you should NOT get undressed... 2556:Remarks to the "Quantum mechanics" section 187:example, would perhaps not be a bad idea. 4962: 4938: 4937: 4925: 4901: 4900: 4888: 4872: 4854: 4823:{\displaystyle \langle x,x\rangle \geq 0} 4797: 4748: 4720: 4692: 4676: 4664: 4290:section to address your second comment. 4022: 4012: 3993: 3987: 3966:. Should this be mentioned? (vn article) 3942: 3926: 3920: 3792: 3154:: there are some aspects in physics that 2990: 1484: 1474: 1463: 1457: 1424: 1411: 1400: 1394: 593: 4388:Also, Timothy, do not forget to sign... 3951:{\displaystyle \bigoplus _{i\in I}H_{i}} 3208:Speaking about bases in the introduction 5009: 2847:) of a quantum mechanical system (...)" 1767:as "unilateral" is deeply misleading. 1493: 634: 2100:potentially infinitely many dimensions 985:File:Completeness in Hilbert space.png 44:Do not edit the contents of this page. 5403:2601:143:480:a4c0:6dd5:ec40:6d14:9d00 5050:taken as an argument but to be funny. 4846:in its second argument, meaning that 312:story about Baez. That would help.-- 7: 4305:That seems to cover it. Good work! ( 4181:Well, supposedly the |p1,p2,...: --> 2774:why replace the correct formulation? 2462:Trivia/popular culture/architecture? 1536:the formal definition. This is the 639:{\displaystyle x-y=x+(-1)\cdot y,\,} 5025:other conventions in a footnote). 3221:article explaining what a base is. 1571:Talk:Group_(mathematics)#Definition 841:Definition section revision request 2862:attempt to make text more readable 2348:countable vector space dimension. 1475: 1412: 24: 5387:Unrelated contrivances in article 1990:Looks like I was beaten too it. 1057:, it is a synonym for "called". 3735:This article never mentions the 2882:... then why not write directly 783:they have the same dimension. - 29: 3964:A course in Functional Analysis 1304:ru:Предгильбертово пространство 496:Rewording "norm of differences" 5203:21:20, 29 September 2019 (UTC) 4943: 4906: 4095:12:32, 30 September 2015 (UTC) 3803: 3001: 2995: 2650:10:26, 25 September 2009 (UTC) 2635:16:51, 24 September 2009 (UTC) 2610:08:53, 23 September 2009 (UTC) 2595:12:29, 11 September 2009 (UTC) 1909:Sławomir Biały: I'm happy you 1905:Inclusion of Folland reference 1317:01:42, 15 September 2009 (UTC) 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review 974:12:44, 11 October 2006 (UTC) 956:07:42, 16 October 2006 (UTC) 942:09:40, 11 October 2006 (UTC) 931:15:03, 10 October 2006 (UTC) 920:15:01, 10 October 2006 (UTC) 909:10:03, 10 October 2006 (UTC) 440:23:05, 16 January 2007 (UTC) 416:00:59, 3 February 2006 (UTC) 406:21:34, 2 February 2006 (UTC) 395:21:15, 2 February 2006 (UTC) 376:06:06, 18 January 2006 (UTC) 333:00:55, 5 November 2005 (UTC) 317:13:02, 28 October 2005 (UTC) 304:02:20, 28 October 2005 (UTC) 289:00:12, 28 October 2005 (UTC) 279:02:20, 28 October 2005 (UTC) 257:02:20, 28 October 2005 (UTC) 4584:Superposition Image Problem 2332:-- but let's be correct. -- 2317:11:51, 31 August 2011 (UTC) 2301:07:33, 31 August 2011 (UTC) 2274:12:52, 30 August 2011 (UTC) 2257:11:16, 30 August 2011 (UTC) 2239:11:00, 30 August 2011 (UTC) 2221:08:56, 30 August 2011 (UTC) 2016:20:26, 5 October 2010 (UTC) 2000:17:33, 5 October 2010 (UTC) 1986:17:29, 5 October 2010 (UTC) 1961:14:26, 5 October 2010 (UTC) 869:23:09, 21 August 2006 (UTC) 794:Hilbert Spaces and Topology 427:01:46, 14 August 2006 (UTC) 5427: 4069:15:07, 12 April 2014 (UTC) 4051:14:26, 12 April 2014 (UTC) 3976:14:03, 12 April 2014 (UTC) 3863:Talk:Quantum teleportation 3713:than arbitrary operators. 3200:06:57, 23 April 2013 (UTC) 3035:11:07, 19 April 2013 (UTC) 2976:10:24, 19 April 2013 (UTC) 2961:09:53, 19 April 2013 (UTC) 2765:13:42, 8 August 2012 (UTC) 2745:16:49, 3 August 2012 (UTC) 2724:16:05, 3 August 2012 (UTC) 2530:14:05, 29 April 2012 (UTC) 2514:13:45, 29 April 2012 (UTC) 2494:10:04, 29 April 2012 (UTC) 2477:03:38, 29 April 2012 (UTC) 2327:Contrast with vector space 2060:15:03, 25 April 2011 (UTC) 2043:14:44, 25 April 2011 (UTC) 1152:also supports me in this. 1021:Taschenbuch der Mathematik 788:08:07, 24 April 2006 (UTC) 744:01:54, 10 March 2006 (UTC) 365:discrete Fourier transform 5411:10:04, 2 April 2020 (UTC) 5170:12:07, 7 March 2017 (UTC) 5102:19:00, 6 March 2017 (UTC) 5070:18:49, 6 March 2017 (UTC) 5037:22:18, 1 March 2017 (UTC) 4999:20:59, 1 March 2017 (UTC) 4626:Inner product's linearity 4625: 4340:are orthogonal if p!=p'.( 3903:02:21, 2 April 2014 (UTC) 3723:05:43, 24 July 2013 (UTC) 3700:14:50, 23 July 2013 (UTC) 3683:14:47, 23 July 2013 (UTC) 3636:16:17, 24 July 2013 (UTC) 3574:15:17, 24 July 2013 (UTC) 3556:14:49, 24 July 2013 (UTC) 3492:14:24, 24 July 2013 (UTC) 3454:13:23, 24 July 2013 (UTC) 3437:11:08, 24 July 2013 (UTC) 3359:00:51, 24 July 2013 (UTC) 3343:16:41, 23 July 2013 (UTC) 3328:14:50, 23 July 2013 (UTC) 3312:14:06, 23 July 2013 (UTC) 3280:07:34, 23 July 2013 (UTC) 3264:15:58, 22 July 2013 (UTC) 3248:15:35, 22 July 2013 (UTC) 3113:does NOT redirect to the 2810: 2693:05:58, 10 June 2012 (UTC) 2677:04:45, 10 June 2012 (UTC) 2123:It just means possibly. 1923:17:59, 24 June 2010 (UTC) 1528:Example before definition 1296:es:Espacio prehilbertiano 830:23:53, 30 June 2006 (UTC) 817:23:24, 30 June 2006 (UTC) 807:22:58, 30 June 2006 (UTC) 727:17:21, 9 March 2006 (UTC) 716:17:07, 9 March 2006 (UTC) 692:04:14, 7 March 2006 (UTC) 675:04:07, 7 March 2006 (UTC) 651:03:52, 7 March 2006 (UTC) 585:03:48, 7 March 2006 (UTC) 572:01:24, 7 March 2006 (UTC) 529:01:15, 7 March 2006 (UTC) 321:A little googling turned 4578:19:13, 10 May 2016 (UTC) 4560:18:37, 10 May 2016 (UTC) 4493:the section on duality. 3007:{\displaystyle \psi (t)} 2185:12:28, 27 May 2011 (UTC) 2165:00:58, 27 May 2011 (UTC) 1897:18:15, 2 June 2010 (UTC) 1878:12:01, 2 June 2010 (UTC) 1851:10:41, 2 June 2010 (UTC) 1810:04:04, 2 June 2010 (UTC) 1779:02:36, 2 June 2010 (UTC) 1747:23:40, 1 June 2010 (UTC) 1724:22:33, 30 May 2010 (UTC) 1701:16:51, 29 May 2010 (UTC) 1646:09:21, 28 May 2010 (UTC) 1598:03:09, 28 May 2010 (UTC) 1564:23:08, 26 May 2010 (UTC) 1450:The net displacement is 1300:it:Spazio prehilbertiano 91:15:17, 1 Mar 2004 (UTC) 5062:Álvaro López de Quadros 4991:Álvaro López de Quadros 4364:concentrated at points 3746:dagger compact category 2712:Hilbert space dimension 2544:13:03, 3 May 2012 (UTC) 2095:17:06, 4 May 2011 (UTC) 1911:included Folland's book 1292:fr:Espace préhilbertien 874:Good Article nomination 5122:be given minimal space 4978: 4824: 4770: 4032: 3952: 3831: 3316:What choice of basis? 3008: 2209:required by convention 1499: 1479: 1440: 1416: 1023:, 7. ed, 2008, p. 678) 640: 369:orthogonal polynomials 5254:, and is named after 5224:complete metric space 4979: 4825: 4771: 4641:The inner product is 4530:Bergman, not Bargmann 4432:Nice. Happy editing. 4083:complex affine spaces 4077:complex affine spaces 4033: 3953: 3832: 3152:demarcation criterion 3009: 2225:Previous discussion: 2197:Bracket (mathematics) 2073:Physics is all gnomes 1500: 1459: 1441: 1396: 983:I uploaded the image 765:comment was added by 641: 42:of past discussions. 5232:absolute convergence 4853: 4796: 4663: 4548:Segal–Bargmann space 4490:closed graph theorem 4482:open mapping theorem 4212:converges absolutely 3986: 3919: 3791: 3158:. And it seems like 3117:article but to the " 2989: 2660:numerical algorithm) 1932:In a pair of edits ( 1675:Rigged Hilbert space 1456: 1393: 915:unnecessary chaff. – 752:What does this mean? 706:frowning on bra-kets 592: 467:comment was added by 5228:mathematical series 4486:Hahn-Banach theorem 4111:converging sequence 3742:no-deleting theorem 3076:" redirects to the 2619:Von Neumann algebra 1945:geometry, whereas " 1836:Group (mathematics) 1546:Group (mathematics) 1184:Just looked at the 1086:source given above 1051:inner product space 455:I've deleted this, 5350:Do not "dumb-down" 5236:norm (mathematics) 4974: 4820: 4766: 4544:Valentine Bargmann 4247:norm (mathematics) 4208:infinite summation 4206:, meaning that if 4028: 4004: 3948: 3937: 3827: 3737:no-cloning theorem 3731:no cloning theorem 3080:article (#redirect 3004: 2901:article (#redirect 2857:of a system (...)" 2831:Hallo there Prof. 1495: 1494: 1436: 1013:sometimes is known 636: 635: 359:17 January 2006). 18:Talk:Hilbert space 5413: 5401:comment added by 5205: 5193:comment added by 5176:Lack of precision 4946: 4909: 4784:positive definite 4606: 4594:comment added by 4526: 4514:comment added by 4403:Physics project. 3989: 3922: 3133:redirects to the 3099:I made a mistake: 3025: 3014:for some instant 2897:redirects to the 2872:"possible states" 2825: 2824: 2803: 2752:Hilbert dimension 2731:Hilbert dimension 2708:Hilbert dimension 2701:Hilbert dimension 2454: 2390: 2168: 2151:comment added by 1972: 1552:the definition. 1369: 1355:comment added by 1288:de:Prähilbertraum 1005:Pre-Hilbert space 778: 481: 104:Quantum Mechanics 81:orthonormal basis 72: 71: 54: 53: 48:current talk page 5418: 5094: 5088:bra-ket notation 5029: 5017: 5014: 4983: 4981: 4980: 4975: 4967: 4966: 4948: 4947: 4939: 4930: 4929: 4911: 4910: 4902: 4893: 4892: 4877: 4876: 4829: 4827: 4826: 4821: 4775: 4773: 4772: 4767: 4753: 4752: 4725: 4724: 4697: 4696: 4681: 4680: 4570: 4288:Separable spaces 4043: 4037: 4035: 4034: 4029: 4027: 4026: 4017: 4016: 4003: 3957: 3955: 3954: 3949: 3947: 3946: 3936: 3836: 3834: 3833: 3828: 3771: 3692: 3566: 3500: 3446: 3351: 3320: 3272: 3197: 3196: 3195: 3183: 3182: 3181: 3047:) and Professor 3023: 3013: 3011: 3010: 3005: 2958: 2957: 2956: 2944: 2943: 2942: 2876:"more precisely" 2840:I have replaced 2821: 2800: 2785: 2779: 2737: 2522: 2486: 2453: 2447: 2440: 2424: 2389: 2383: 2376: 2309: 2249: 2177: 2167: 2145: 2127: 2052: 1967: 1953: 1843: 1771: 1556: 1542:Knowledge is not 1514: 1504: 1502: 1501: 1496: 1489: 1488: 1478: 1473: 1445: 1443: 1442: 1437: 1429: 1428: 1415: 1410: 1368: 1349: 1315: 1313: 1249: 1218: 1216: 1156: 1122: 1099: 1097: 1061: 1039: 1037: 992: 760: 645: 643: 642: 637: 460: 63: 56: 55: 33: 32: 26: 5426: 5425: 5421: 5420: 5419: 5417: 5416: 5415: 5389: 5216:Euclidean space 5211: 5178: 5119:be very visible 5092: 5027: 5021: 5020: 5015: 5011: 4958: 4921: 4884: 4868: 4851: 4850: 4794: 4793: 4744: 4716: 4688: 4672: 4661: 4660: 4628: 4586: 4568: 4552:Boris Tsirelson 4532: 4467: 4434:Boris Tsirelson 4405:Boris Tsirelson 4390:Boris Tsirelson 4376:Boris Tsirelson 4323:Boris Tsirelson 4169:Boris Tsirelson 4148:Boris Tsirelson 4102: 4079: 4041: 4018: 4008: 3984: 3983: 3938: 3917: 3916: 3910: 3891: 3840:Boris Tsirelson 3789: 3788: 3769: 3733: 3715:Boris Tsirelson 3690: 3564: 3498: 3444: 3349: 3318: 3270: 3256:Boris Tsirelson 3210: 3193: 3192: 3187: 3186: 3177: 3176: 3169: 3168: 3115:"quantum state" 3078:"quantum state" 3027:Boris Tsirelson 2987: 2986: 2954: 2953: 2948: 2947: 2938: 2937: 2930: 2929: 2899:"quantum state" 2811: 2796: 2777: 2735: 2705: 2685:Boris Tsirelson 2669:David R. Ingham 2627:Boris Tsirelson 2587:Boris Tsirelson 2558: 2520: 2484: 2464: 2443: 2441: 2422: 2379: 2377: 2329: 2307: 2247: 2193: 2175: 2146: 2125: 2110:Boris Tsirelson 2102: 2068: 2050: 2028: 1951: 1942:Euclidean space 1930: 1928:Euclidean space 1907: 1870:Jakob.scholbach 1841: 1769: 1554: 1530: 1512: 1480: 1454: 1453: 1420: 1391: 1390: 1386: 1379: 1350: 1337: 1309: 1307: 1247: 1212: 1210: 1154: 1148:Actually, your 1120: 1093: 1091: 1059: 1033: 1031: 1007: 990: 981: 966: 876: 843: 796: 761:—The preceding 754: 708: 590: 589: 566: 557: 548: 542: 498: 450: 344: 238: 229: 156:Kähler manifold 77: 59: 30: 22: 21: 20: 12: 11: 5: 5424: 5422: 5388: 5385: 5374:Sławomir Biały 5370: 5369: 5368: 5367: 5366: 5365: 5355:Sławomir Biały 5336: 5335: 5334: 5333: 5315: 5314: 5304:Sławomir Biały 5272: 5271: 5210: 5207: 5195:217.95.164.135 5177: 5174: 5173: 5172: 5150: 5135: 5132: 5125: 5124: 5123: 5120: 5109: 5108: 5107: 5106: 5105: 5104: 5093:Sławomir Biały 5075: 5074: 5073: 5072: 5054: 5053: 5052: 5051: 5047: 5040: 5039: 5028:Sławomir Biały 5019: 5018: 5008: 5007: 4987: 4985: 4984: 4973: 4970: 4965: 4961: 4957: 4954: 4951: 4945: 4942: 4936: 4933: 4928: 4924: 4920: 4917: 4914: 4908: 4905: 4899: 4896: 4891: 4887: 4883: 4880: 4875: 4871: 4867: 4864: 4861: 4858: 4840: 4839: 4832: 4831: 4830: 4819: 4816: 4813: 4810: 4807: 4804: 4801: 4788: 4787: 4779: 4778: 4777: 4776: 4765: 4762: 4759: 4756: 4751: 4747: 4743: 4740: 4737: 4734: 4731: 4728: 4723: 4719: 4715: 4712: 4709: 4706: 4703: 4700: 4695: 4691: 4687: 4684: 4679: 4675: 4671: 4668: 4655: 4654: 4637: 4627: 4624: 4623: 4622: 4585: 4582: 4581: 4580: 4569:Sławomir Biały 4540:Bergman kernel 4536:Stefan Bergman 4531: 4528: 4506: 4505: 4495:Sławomir Biały 4463: 4451: 4450: 4449: 4448: 4447: 4446: 4445: 4444: 4400: 4386: 4354: 4353: 4319: 4318: 4292:Sławomir Biały 4284: 4283: 4282: 4281: 4280: 4279: 4278: 4277: 4251:Sławomir Biały 4216:Sławomir Biały 4199: 4198: 4197: 4196: 4159: 4158: 4128: 4127: 4123: 4122: 4116: 4115: 4101: 4098: 4078: 4075: 4074: 4073: 4072: 4071: 4054: 4053: 4042:Sławomir Biały 4025: 4021: 4015: 4011: 4007: 4002: 3999: 3996: 3992: 3959: 3958: 3945: 3941: 3935: 3932: 3929: 3925: 3909: 3906: 3890: 3887: 3886: 3885: 3884: 3883: 3882: 3881: 3853: 3852: 3851: 3850: 3826: 3823: 3820: 3817: 3814: 3811: 3808: 3805: 3802: 3799: 3796: 3782: 3781: 3770:Sławomir Biały 3732: 3729: 3728: 3727: 3726: 3725: 3691:Sławomir Biały 3663: 3662: 3661: 3660: 3659: 3658: 3657: 3656: 3655: 3654: 3653: 3652: 3651: 3650: 3649: 3648: 3647: 3646: 3645: 3644: 3643: 3642: 3641: 3640: 3639: 3638: 3599: 3598: 3597: 3596: 3595: 3594: 3593: 3592: 3591: 3590: 3589: 3588: 3587: 3586: 3585: 3584: 3583: 3582: 3581: 3580: 3579: 3578: 3577: 3576: 3565:Sławomir Biały 3524: 3523: 3522: 3521: 3520: 3519: 3518: 3517: 3516: 3515: 3514: 3513: 3512: 3511: 3510: 3509: 3508: 3507: 3499:Sławomir Biały 3476: 3475: 3474: 3464: 3460: 3445:Sławomir Biały 3413: 3412: 3411: 3410: 3409: 3408: 3407: 3406: 3405: 3404: 3403: 3402: 3401: 3400: 3399: 3398: 3380: 3379: 3378: 3377: 3376: 3375: 3374: 3373: 3372: 3371: 3370: 3369: 3368: 3367: 3350:Sławomir Biały 3319:Sławomir Biały 3301: 3297: 3293: 3289: 3283: 3282: 3271:Sławomir Biały 3266: 3234: 3233: 3218: 3217: 3209: 3206: 3205: 3204: 3203: 3202: 3188: 3170: 3156:sound very odd 3148: 3147: 3145: 3144: 3143: 3142: 3140: 3122: 3119:Density matrix 3116: 3109: 3100: 3097: 3096: 3095: 3094: 3091: 3079: 3071: 3064: 3061: 3060: 3057: 3056: 3003: 3000: 2997: 2994: 2983: 2979: 2978: 2949: 2931: 2927: 2925: 2923: 2920: 2919: 2918: 2917: 2916: 2915: 2914: 2912: 2900: 2896: 2887: 2885: 2881: 2877: 2873: 2865: 2859: 2855:quantum states 2849: 2836: 2823: 2822: 2808: 2807: 2804: 2794: 2791: 2783: 2776: 2771: 2770: 2769: 2768: 2767: 2736:Sławomir Biały 2704: 2703:a common term? 2697: 2696: 2695: 2656: 2655: 2654: 2653: 2622: 2557: 2554: 2553: 2552: 2551: 2550: 2549: 2548: 2547: 2546: 2521:Sławomir Biały 2502:paradox, than 2485:Sławomir Biały 2463: 2460: 2459: 2458: 2435: 2434: 2423:Sławomir Biały 2417: 2416: 2415: 2414: 2397: 2396: 2395: 2394: 2374: 2370: 2367: 2361: 2360: 2328: 2325: 2324: 2323: 2322: 2321: 2320: 2319: 2308:Sławomir Biały 2277: 2276: 2260: 2259: 2248:Sławomir Biały 2192: 2191:Angle brackets 2189: 2188: 2187: 2176:Sławomir Biały 2138: 2137: 2126:Sławomir Biały 2101: 2098: 2067: 2066:Good work here 2064: 2063: 2062: 2051:Sławomir Biały 2027: 2024: 2023: 2022: 2021: 2020: 2019: 2018: 1952:Sławomir Biały 1929: 1926: 1906: 1903: 1902: 1901: 1900: 1899: 1881: 1880: 1864: 1863: 1862: 1861: 1860: 1859: 1858: 1857: 1856: 1855: 1854: 1853: 1842:Sławomir Biały 1821: 1820: 1819: 1818: 1817: 1816: 1815: 1814: 1813: 1812: 1788: 1787: 1786: 1785: 1784: 1783: 1782: 1781: 1770:Sławomir Biały 1754: 1753: 1752: 1751: 1750: 1749: 1729: 1728: 1727: 1726: 1708: 1707: 1706: 1705: 1704: 1703: 1684: 1683: 1682: 1681: 1680: 1679: 1665: 1664: 1663: 1662: 1661: 1660: 1651: 1650: 1649: 1648: 1631: 1630: 1629: 1628: 1621: 1620: 1619: 1618: 1611: 1610: 1609: 1608: 1601: 1600: 1585:WP:NOTTEXTBOOK 1581: 1580: 1575: 1574: 1555:Sławomir Biały 1529: 1526: 1525: 1524: 1513:Sławomir Biały 1507: 1506: 1505: 1492: 1487: 1483: 1477: 1472: 1469: 1466: 1462: 1448: 1447: 1446: 1435: 1432: 1427: 1423: 1419: 1414: 1409: 1406: 1403: 1399: 1384: 1377: 1357:157.193.53.246 1336: 1333: 1332: 1331: 1330: 1329: 1328: 1327: 1326: 1325: 1324: 1323: 1322: 1321: 1320: 1319: 1270: 1269: 1268: 1267: 1266: 1265: 1264: 1263: 1262: 1261: 1260: 1259: 1248:Sławomir Biały 1231: 1230: 1229: 1228: 1227: 1226: 1225: 1224: 1223: 1222: 1207: 1206: 1205: 1202: 1173: 1172: 1171: 1170: 1169: 1168: 1167: 1166: 1155:Sławomir Biały 1139: 1138: 1137: 1136: 1135: 1134: 1133: 1132: 1121:Sławomir Biały 1108: 1107: 1106: 1105: 1104: 1103: 1074: 1073: 1072: 1071: 1060:Sławomir Biały 1044: 1043: 1024: 1006: 1003: 991:Sławomir Biały 980: 977: 965: 962: 961: 960: 959: 958: 945: 944: 923: 922: 902: 901: 898: 895: 892: 888: 885: 875: 872: 862: 861: 857: 856: 851: 850: 842: 839: 837: 835: 834: 833: 832: 820: 819: 795: 792: 791: 790: 753: 750: 749: 748: 747: 746: 730: 729: 707: 704: 703: 702: 701: 700: 699: 698: 697: 696: 695: 694: 660: 658: 657: 656: 655: 654: 653: 633: 630: 627: 624: 621: 618: 615: 612: 609: 606: 603: 600: 597: 575: 574: 562: 553: 544: 538: 497: 494: 493: 492: 463:The preceding 449: 446: 445: 444: 443: 442: 419: 418: 398: 397: 379: 378: 357:User:Sayfadeen 343: 340: 339: 338: 337: 336: 335: 334: 306: 305: 281: 280: 259: 258: 237: 234: 228: 225: 224: 223: 219: 218: 217: 216: 215: 214: 213: 212: 198: 197: 196: 195: 194: 193: 192: 191: 164: 163: 162: 161: 160: 159: 139: 138: 137: 136: 129: 128: 120: 119: 99: 98: 97: 76: 73: 70: 69: 64: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 5423: 5414: 5412: 5408: 5404: 5400: 5393: 5386: 5384: 5383: 5379: 5375: 5364: 5360: 5356: 5351: 5347: 5342: 5341: 5340: 5339: 5338: 5337: 5332: 5328: 5324: 5319: 5318: 5317: 5316: 5313: 5309: 5305: 5300: 5296: 5292: 5288: 5287: 5286: 5285: 5281: 5277: 5269: 5265: 5261: 5257: 5256:David Hilbert 5253: 5252:vector spaces 5249: 5245: 5244:Hilbert space 5241: 5240: 5239: 5237: 5233: 5229: 5225: 5221: 5217: 5208: 5206: 5204: 5200: 5196: 5192: 5184: 5181: 5175: 5171: 5167: 5163: 5159: 5158:inner product 5156:should be in 5155: 5151: 5148: 5143: 5139: 5136: 5133: 5130: 5126: 5121: 5118: 5117: 5115: 5111: 5110: 5103: 5099: 5095: 5089: 5085: 5081: 5080: 5079: 5078: 5077: 5076: 5071: 5067: 5063: 5058: 5057: 5056: 5055: 5048: 5044: 5043: 5042: 5041: 5038: 5034: 5030: 5023: 5022: 5013: 5010: 5006: 5005: 5004: 5000: 4996: 4992: 4988: 4971: 4963: 4959: 4955: 4952: 4940: 4934: 4926: 4922: 4918: 4915: 4903: 4897: 4889: 4885: 4881: 4878: 4873: 4869: 4865: 4862: 4859: 4849: 4848: 4847: 4845: 4837: 4833: 4817: 4814: 4808: 4805: 4802: 4792: 4791: 4790: 4789: 4785: 4781: 4780: 4763: 4757: 4754: 4749: 4745: 4738: 4735: 4729: 4726: 4721: 4717: 4710: 4707: 4701: 4698: 4693: 4689: 4685: 4682: 4677: 4673: 4669: 4659: 4658: 4657: 4656: 4652: 4648: 4644: 4640: 4639: 4638: 4635: 4632: 4621: 4617: 4613: 4609: 4608: 4607: 4605: 4601: 4597: 4596:64.180.21.136 4593: 4583: 4579: 4575: 4571: 4564: 4563: 4562: 4561: 4557: 4553: 4549: 4545: 4541: 4537: 4529: 4527: 4525: 4521: 4517: 4516:75.163.175.56 4513: 4504: 4500: 4496: 4491: 4487: 4483: 4478: 4477: 4476: 4475: 4471: 4466: 4461: 4457: 4456:Banach spaces 4443: 4439: 4435: 4431: 4430: 4428: 4424: 4420: 4416: 4415: 4414: 4410: 4406: 4401: 4399: 4395: 4391: 4387: 4385: 4381: 4377: 4371: 4367: 4362: 4358: 4357: 4356: 4355: 4351: 4347: 4343: 4335: 4334: 4333: 4332: 4328: 4324: 4316: 4312: 4308: 4304: 4303: 4302: 4301: 4297: 4293: 4289: 4275: 4271: 4267: 4262: 4261: 4260: 4256: 4252: 4248: 4243: 4242: 4240: 4236: 4232: 4227: 4226: 4225: 4221: 4217: 4213: 4209: 4205: 4201: 4200: 4194: 4190: 4186: 4180: 4179: 4178: 4174: 4170: 4165: 4161: 4160: 4157: 4153: 4149: 4145: 4144: 4143: 4141: 4137: 4133: 4125: 4124: 4118: 4117: 4112: 4107: 4106: 4105: 4099: 4097: 4096: 4092: 4088: 4084: 4076: 4070: 4066: 4062: 4058: 4057: 4056: 4055: 4052: 4048: 4044: 4023: 4013: 4009: 4000: 3997: 3994: 3990: 3980: 3979: 3978: 3977: 3973: 3969: 3965: 3943: 3939: 3933: 3930: 3927: 3923: 3915: 3914: 3913: 3907: 3905: 3904: 3900: 3896: 3888: 3880: 3876: 3872: 3868: 3864: 3859: 3858: 3857: 3856: 3855: 3854: 3849: 3845: 3841: 3824: 3821: 3818: 3815: 3812: 3809: 3806: 3800: 3797: 3794: 3786: 3785: 3784: 3783: 3780: 3776: 3772: 3766: 3765: 3764: 3763: 3759: 3755: 3749: 3747: 3743: 3738: 3730: 3724: 3720: 3716: 3712: 3708: 3703: 3702: 3701: 3697: 3693: 3687: 3686: 3685: 3684: 3680: 3676: 3672: 3668: 3637: 3633: 3629: 3625: 3624: 3623: 3622: 3621: 3620: 3619: 3618: 3617: 3616: 3615: 3614: 3613: 3612: 3611: 3610: 3609: 3608: 3607: 3606: 3605: 3604: 3603: 3602: 3601: 3600: 3575: 3571: 3567: 3561: 3560: 3559: 3558: 3557: 3553: 3549: 3544: 3543: 3542: 3541: 3540: 3539: 3538: 3537: 3536: 3535: 3534: 3533: 3532: 3531: 3530: 3529: 3528: 3527: 3526: 3525: 3505: 3501: 3495: 3494: 3493: 3489: 3485: 3481: 3477: 3472: 3471: 3469: 3465: 3461: 3457: 3456: 3455: 3451: 3447: 3440: 3439: 3438: 3434: 3430: 3425: 3424: 3423: 3422: 3421: 3420: 3419: 3418: 3417: 3416: 3415: 3414: 3396: 3395: 3394: 3393: 3392: 3391: 3390: 3389: 3388: 3387: 3386: 3385: 3384: 3383: 3382: 3381: 3365: 3364: 3362: 3361: 3360: 3356: 3352: 3346: 3345: 3344: 3340: 3336: 3331: 3330: 3329: 3325: 3321: 3315: 3314: 3313: 3309: 3305: 3302: 3298: 3294: 3290: 3287: 3286: 3285: 3284: 3281: 3277: 3273: 3267: 3265: 3261: 3257: 3252: 3251: 3250: 3249: 3245: 3241: 3237: 3230: 3229: 3228: 3225: 3222: 3215: 3214: 3213: 3207: 3201: 3198: 3191: 3184: 3180: 3175: 3174: 3165: 3161: 3157: 3153: 3149: 3146: 3138: 3137:quantum state 3134: 3132: 3128: 3124: 3123: 3120: 3114: 3112: 3107: 3103: 3102: 3101: 3098: 3093: 3089: 3088:quantum state 3085: 3083: 3082:quantum state 3077: 3075: 3069: 3066: 3065: 3062: 3058: 3054: 3050: 3046: 3042: 3038: 3037: 3036: 3032: 3028: 3021: 3017: 2998: 2992: 2984: 2981: 2980: 2977: 2973: 2969: 2965: 2964: 2963: 2962: 2959: 2952: 2945: 2941: 2936: 2935: 2910: 2909:quantum state 2906: 2904: 2903:quantum state 2898: 2894: 2890: 2888: 2884:"pure states" 2883: 2880:"pure states" 2879: 2878:written with 2875: 2871: 2869: 2868: 2867: 2866: 2863: 2858: 2856: 2851: 2848: 2846: 2841: 2839: 2838: 2837: 2834: 2829: 2828: 2819: 2817: 2816:Hilbert space 2809: 2802: 2801: 2799: 2789: 2788:Occam's razor 2784: 2780: 2775: 2772: 2766: 2762: 2758: 2754: 2753: 2748: 2747: 2746: 2742: 2738: 2732: 2728: 2727: 2726: 2725: 2721: 2717: 2713: 2709: 2702: 2698: 2694: 2690: 2686: 2681: 2680: 2679: 2678: 2674: 2670: 2665: 2661: 2651: 2647: 2643: 2638: 2637: 2636: 2632: 2628: 2623: 2620: 2615: 2614: 2613: 2611: 2607: 2603: 2597: 2596: 2592: 2588: 2584: 2581: 2579: 2575: 2570: 2567: 2564: 2561: 2555: 2545: 2541: 2537: 2533: 2532: 2531: 2527: 2523: 2517: 2516: 2515: 2512: 2509: 2505: 2504:Hilbert space 2501: 2500:Hilbert hotel 2497: 2496: 2495: 2491: 2487: 2481: 2480: 2479: 2478: 2474: 2470: 2461: 2457: 2452: 2448: 2446: 2437: 2436: 2433: 2429: 2425: 2419: 2418: 2413: 2409: 2405: 2401: 2400: 2399: 2398: 2393: 2388: 2384: 2382: 2375: 2371: 2368: 2365: 2364: 2363: 2362: 2359: 2355: 2351: 2346: 2345: 2344: 2343: 2339: 2335: 2326: 2318: 2314: 2310: 2304: 2303: 2302: 2298: 2294: 2290: 2285: 2281: 2280: 2279: 2278: 2275: 2271: 2267: 2264:characters.-- 2262: 2261: 2258: 2254: 2250: 2243: 2242: 2241: 2240: 2236: 2232: 2228: 2223: 2222: 2218: 2214: 2210: 2205: 2202: 2198: 2190: 2186: 2182: 2178: 2171: 2170: 2169: 2166: 2162: 2158: 2154: 2150: 2142: 2136: 2132: 2128: 2122: 2121: 2120: 2119: 2115: 2111: 2107: 2099: 2097: 2096: 2092: 2088: 2087:78.86.197.227 2083: 2082: 2078: 2074: 2065: 2061: 2057: 2053: 2047: 2046: 2045: 2044: 2040: 2036: 2032: 2026:Notes-Section 2025: 2017: 2014: 2011: 2007: 2003: 2002: 2001: 1997: 1993: 1989: 1988: 1987: 1983: 1979: 1976: 1970: 1965: 1964: 1963: 1962: 1958: 1954: 1948: 1943: 1939: 1935: 1927: 1925: 1924: 1920: 1916: 1912: 1904: 1898: 1894: 1890: 1885: 1884: 1883: 1882: 1879: 1875: 1871: 1866: 1865: 1852: 1848: 1844: 1837: 1833: 1832: 1831: 1830: 1829: 1828: 1827: 1826: 1825: 1824: 1823: 1822: 1811: 1807: 1803: 1798: 1797: 1796: 1795: 1794: 1793: 1792: 1791: 1790: 1789: 1780: 1776: 1772: 1766: 1762: 1761: 1760: 1759: 1758: 1757: 1756: 1755: 1748: 1744: 1740: 1735: 1734: 1733: 1732: 1731: 1730: 1725: 1721: 1717: 1712: 1711: 1710: 1709: 1702: 1698: 1694: 1690: 1689: 1688: 1687: 1686: 1685: 1676: 1671: 1670: 1669: 1668: 1667: 1666: 1657: 1656: 1655: 1654: 1653: 1652: 1647: 1643: 1639: 1635: 1634: 1633: 1632: 1625: 1624: 1623: 1622: 1615: 1614: 1613: 1612: 1605: 1604: 1603: 1602: 1599: 1595: 1591: 1586: 1583: 1582: 1577: 1576: 1572: 1568: 1567: 1566: 1565: 1561: 1557: 1551: 1547: 1543: 1539: 1535: 1527: 1523: 1519: 1515: 1508: 1490: 1485: 1481: 1470: 1467: 1464: 1460: 1452: 1451: 1449: 1433: 1425: 1421: 1407: 1404: 1401: 1397: 1389: 1388: 1383: 1376: 1372: 1371: 1370: 1366: 1362: 1358: 1354: 1346: 1343: 1340: 1334: 1318: 1314: 1312: 1305: 1301: 1297: 1293: 1289: 1284: 1283: 1282: 1281: 1280: 1279: 1278: 1277: 1276: 1275: 1274: 1273: 1272: 1271: 1258: 1254: 1250: 1243: 1242: 1241: 1240: 1239: 1238: 1237: 1236: 1235: 1234: 1233: 1232: 1221: 1217: 1215: 1208: 1203: 1200: 1199: 1196: 1192: 1187: 1183: 1182: 1181: 1180: 1179: 1178: 1177: 1176: 1175: 1174: 1165: 1161: 1157: 1151: 1147: 1146: 1145: 1144: 1143: 1142: 1141: 1140: 1131: 1127: 1123: 1116: 1115: 1114: 1113: 1112: 1111: 1110: 1109: 1102: 1098: 1096: 1089: 1085: 1080: 1079: 1078: 1077: 1076: 1075: 1070: 1066: 1062: 1056: 1052: 1048: 1047: 1046: 1045: 1042: 1038: 1036: 1029: 1025: 1022: 1018: 1017: 1016: 1014: 1009: 1004: 1002: 1001: 997: 993: 986: 979:Image problem 978: 976: 975: 972: 963: 957: 954: 949: 948: 947: 946: 943: 940: 935: 934: 933: 932: 929: 921: 918: 913: 912: 911: 910: 907: 899: 896: 893: 889: 886: 884:introduction. 882: 881: 880: 873: 871: 870: 867: 859: 858: 853: 852: 848: 847: 846: 840: 838: 831: 828: 824: 823: 822: 821: 818: 815: 811: 810: 809: 808: 805: 804:68.98.221.241 800: 793: 789: 786: 781: 780: 779: 776: 772: 768: 767:59.144.16.174 764: 757: 751: 745: 742: 738: 734: 733: 732: 731: 728: 725: 720: 719: 718: 717: 714: 705: 693: 690: 686: 682: 678: 677: 676: 673: 669: 668: 667: 666: 665: 664: 663: 662: 661: 652: 649: 631: 628: 625: 619: 616: 610: 607: 604: 601: 598: 595: 588: 587: 586: 583: 579: 578: 577: 576: 573: 570: 565: 561: 556: 552: 547: 541: 537: 533: 532: 531: 530: 527: 521: 519: 513: 511: 507: 501: 495: 491: 488: 484: 483: 482: 479: 475: 471: 468: 466: 458: 453: 448:this is wrong 447: 441: 438: 433: 432: 431: 430: 429: 428: 425: 424:69.44.253.177 417: 414: 410: 409: 408: 407: 404: 396: 393: 389: 388: 387: 385: 377: 374: 370: 366: 362: 361: 360: 358: 352: 348: 342:The Layperson 341: 332: 328: 324: 320: 319: 318: 315: 310: 309: 308: 307: 303: 299: 294: 293: 292: 290: 287: 278: 274: 269: 268: 267: 263: 256: 252: 248: 244: 243: 242: 233: 226: 221: 220: 211: 206: 205: 204: 203: 202: 201: 200: 199: 190: 185: 184: 183: 179: 174: 173:hilbert space 170: 169: 168: 167: 166: 165: 157: 153: 152:monster group 149: 148:monster group 145: 144: 143: 142: 141: 140: 133: 132: 131: 130: 127: 122: 121: 116: 115: 114: 112: 107: 105: 94: 93: 92: 90: 84: 82: 74: 68: 65: 62: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 5397:— Preceding 5394: 5390: 5371: 5349: 5273: 5243: 5220:vector space 5212: 5189:— Preceding 5185: 5182: 5179: 5153: 5146: 5141: 5128: 5116:, it should 5113: 5083: 5012: 5002: 5001: 4989: 4986: 4841: 4835: 4650: 4646: 4636: 4633: 4629: 4590:— Preceding 4587: 4533: 4510:— Preceding 4507: 4452: 4369: 4365: 4360: 4339:and |p': --> 4320: 4287: 4285: 4163: 4129: 4103: 4080: 3963: 3960: 3911: 3892: 3750: 3734: 3710: 3706: 3667:self-adjoint 3664: 3238: 3235: 3226: 3223: 3219: 3211: 3189: 3178: 3172: 3171: 3130: 3110: 3067: 3049:B. Tsirelson 3039:Hallo there 3019: 3015: 2950: 2939: 2933: 2932: 2921: 2852: 2842: 2833:B. Tsirelson 2830: 2827: 2826: 2813: 2795: 2792: 2750: 2711: 2707: 2706: 2700: 2664:Banach space 2657: 2598: 2585: 2582: 2578:general case 2573: 2571: 2568: 2565: 2562: 2559: 2465: 2444: 2380: 2330: 2288: 2284:User:Slawekb 2224: 2206: 2200: 2194: 2153:Mathview2011 2147:— Preceding 2143: 2139: 2103: 2084: 2069: 2029: 2010:Arthur Rubin 1974: 1946: 1940:), the term 1931: 1908: 1549: 1537: 1533: 1531: 1381: 1374: 1347: 1344: 1341: 1338: 1310: 1213: 1190: 1185: 1149: 1094: 1083: 1050: 1034: 1028:semantically 1027: 1020: 1012: 1010: 1008: 982: 967: 924: 903: 877: 863: 844: 836: 801: 797: 758: 755: 709: 684: 680: 659: 563: 559: 554: 550: 545: 539: 535: 522: 517: 514: 509: 505: 502: 499: 462: 454: 451: 437:Sojourner001 420: 399: 380: 353: 349: 345: 286:JackSarfatti 282: 264: 260: 246: 239: 230: 108: 100: 89:Jitse Niesen 85: 78: 60: 43: 37: 5289:Nowhere in 5268:engineering 5260:mathematics 5160:, not here. 5112:As it is a 4612:Steelpillow 4419:TimothyRias 4342:TimothyRias 4307:TimothyRias 4266:TimothyRias 4231:TimothyRias 4210:of vectors 4185:TimothyRias 4132:TimothyRias 4100:GA comments 4087:Steelpillow 3675:Incnis Mrsi 3106:pure states 3074:pure states 3018:; they are 2893:pure states 2853:"(...) the 2845:pure states 2786:"(...) The 2642:TimothyRias 2602:TimothyRias 1915:Brews ohare 1638:TimothyRias 1351:—Preceding 1191:Taschenbuch 1186:Taschenbuch 1150:Taschenbuch 953:RelHistBuff 939:RelHistBuff 906:RelHistBuff 737:trace class 411:Go lethe! 386:2 feb 2006 227:von Neumann 36:This is an 5129:rationales 5114:convention 4844:antilinear 4470:WP Physics 3908:Direct sum 3871:User:Linas 3754:User:Linas 3478:Also from 3166:) Cheers. 3131:(singular) 3127:pure state 3121:" article. 3068:point # 2 1936:and again 1030:correct). 964:Definition 713:Wrath0fb0b 549:is that || 367:, and the 5291:WP:JARGON 5140:notation 5084:at length 5003:Footnotes 4114:solution. 3232:spectrum. 3125:Anyway " 2757:Chricho ∀ 2716:Chricho ∀ 2451:Wolfowitz 2387:Wolfowitz 2366:Liuyipei, 1992:Thenub314 1978:Thenub314 1969:Thenub314 1084:Bronstein 971:Thenub314 210:Sayfadeen 67:Archive 2 61:Archive 1 5399:unsigned 5191:unsigned 4592:unsigned 4512:unsigned 4465:κοντριβς 4460:Headbomb 4204:complete 3671:spectrum 3466:Quoting 3194:arbonaro 3111:(plural) 3070:(And if 3020:possible 2955:arbonaro 2536:Kdammers 2469:Kdammers 2373:numbers? 2334:Liuyipei 2201:must not 2161:contribs 2149:unsigned 2035:Vilietha 1538:opposite 1365:contribs 1353:unsigned 775:contribs 763:unsigned 478:contribs 470:Scineram 465:unsigned 189:Passw0rd 75:Old talk 5346:WP:MTAA 5323:Widsith 5299:WP:LEAD 5295:WP:MTAA 5276:Widsith 5264:physics 5248:vectors 5162:YohanN7 5138:Bra-ket 4338:|p: --> 4081:How do 4061:YohanN7 3968:YohanN7 3895:Dratman 3711:notable 3707:notable 3628:Elferdo 3548:Elferdo 3484:Elferdo 3429:Elferdo 3335:Elferdo 3304:Elferdo 3240:Elferdo 3141:article 3041:Mct mht 2968:Mct mht 2926:Cheers. 2889:And if 2874:can be 2820:article 2404:Mct mht 2350:Mct mht 2293:bungalo 2289:do have 2213:bungalo 2006:WT:MATH 1055:Wordnet 866:DrEricH 384:pgt2006 39:archive 5266:, and 5234:, and 5209:Jargon 5154:really 4643:linear 4542:, not 3179:aurice 3092:then?) 2940:aurice 2860:in an 2850:with 2749:Done: 2445:Kiefer 2381:Kiefer 2013:(talk) 1889:RDBury 1802:Tercer 1739:Tercer 1716:Arcfrk 1693:Tercer 1590:Tercer 1550:before 1534:before 1311:Cs32en 1214:Cs32en 1095:Cs32en 1035:Cs32en 672:Loodog 582:Loodog 526:Loodog 5297:, or 5142:would 4372:: --> 4363:: --> 3300:that. 2913:then? 2574:ideal 2266:LutzL 2231:LutzL 2008:. — 827:lethe 814:lethe 785:lethe 741:linas 724:lethe 689:lethe 648:CSTAR 569:lethe 487:CSTAR 413:linas 403:lethe 392:lethe 373:linas 327:Lethe 314:CSTAR 298:Lethe 273:Lethe 251:Lethe 178:Lethe 126:Lethe 16:< 5407:talk 5378:talk 5359:talk 5327:talk 5308:talk 5280:talk 5199:talk 5166:talk 5147:same 5098:talk 5066:talk 5033:talk 4995:talk 4838:= 0. 4649:and 4616:Talk 4600:talk 4574:talk 4556:talk 4546:and 4538:and 4534:See 4520:talk 4499:talk 4438:talk 4423:talk 4409:talk 4394:talk 4380:talk 4346:talk 4327:talk 4311:talk 4296:talk 4270:talk 4255:talk 4235:talk 4220:talk 4189:talk 4173:talk 4164:free 4152:talk 4136:talk 4091:Talk 4065:talk 4047:talk 3982:sum 3972:talk 3899:talk 3875:talk 3867:POVM 3844:talk 3775:talk 3758:talk 3719:talk 3696:talk 3679:talk 3632:talk 3570:talk 3552:talk 3504:talk 3488:talk 3450:talk 3433:talk 3355:talk 3339:talk 3324:talk 3308:talk 3276:talk 3260:talk 3244:talk 3053:talk 3045:talk 3031:talk 2972:talk 2761:talk 2755:. -- 2741:talk 2720:talk 2689:talk 2673:talk 2646:talk 2631:talk 2606:talk 2591:talk 2540:talk 2526:talk 2490:talk 2473:talk 2428:talk 2408:talk 2354:talk 2338:talk 2313:talk 2297:talk 2270:talk 2253:talk 2235:talk 2217:talk 2181:talk 2157:talk 2131:talk 2114:talk 2091:talk 2077:talk 2056:talk 2039:talk 1996:talk 1982:talk 1973:). 1957:talk 1938:here 1934:here 1919:talk 1893:talk 1874:talk 1847:talk 1806:talk 1775:talk 1743:talk 1720:talk 1697:talk 1678:yet! 1642:talk 1594:talk 1560:talk 1518:talk 1361:talk 1253:talk 1195:norm 1160:talk 1126:talk 1065:talk 996:talk 928:Joke 917:Joke 771:talk 512:. 474:talk 331:Talk 323:this 302:Talk 277:Talk 255:Talk 182:Talk 171:The 111:here 4337:--> 4120:--> 2870:If 2699:Is 777:) . 524:of. 480:) . 247:and 5409:) 5380:) 5361:) 5348:: 5329:) 5310:) 5293:, 5282:) 5262:, 5242:A 5230:, 5226:, 5222:, 5218:, 5201:) 5168:) 5100:) 5068:) 5035:) 4997:) 4969:⟩ 4950:⟨ 4944:¯ 4932:⟩ 4913:⟨ 4907:¯ 4895:⟩ 4857:⟨ 4815:≥ 4812:⟩ 4800:⟨ 4761:⟩ 4742:⟨ 4733:⟩ 4714:⟨ 4705:⟩ 4667:⟨ 4618:) 4602:) 4576:) 4558:) 4550:. 4522:) 4501:) 4472:} 4468:– 4440:) 4429:) 4425:) 4411:) 4396:) 4382:) 4348:) 4329:) 4313:) 4298:) 4272:) 4257:) 4241:) 4237:) 4222:) 4191:) 4175:) 4154:) 4142:) 4138:) 4093:) 4067:) 4049:) 4020:‖ 4006:‖ 3998:∈ 3991:∑ 3974:) 3931:∈ 3924:⨁ 3901:) 3877:) 3846:) 3822:⊗ 3816:∈ 3810:⊗ 3804:↦ 3798:∋ 3777:) 3760:) 3721:) 3698:) 3681:) 3634:) 3572:) 3554:) 3490:) 3470:: 3452:) 3435:) 3357:) 3341:) 3326:) 3310:) 3278:) 3262:) 3246:) 3129:" 3055:), 3033:) 2993:ψ 2974:) 2812:— 2806:” 2782:“ 2763:) 2743:) 2722:) 2691:) 2675:) 2648:) 2633:) 2612:) 2608:) 2593:) 2542:) 2528:) 2492:) 2475:) 2430:) 2410:) 2356:) 2340:) 2315:) 2299:) 2272:) 2255:) 2237:) 2229:-- 2219:) 2211:. 2183:) 2163:) 2159:• 2133:) 2116:) 2108:? 2093:) 2079:) 2071:-- 2058:) 2041:) 1998:) 1984:) 1959:) 1921:) 1895:) 1876:) 1849:) 1808:) 1800:-- 1777:) 1745:) 1722:) 1699:) 1644:) 1596:) 1562:) 1520:) 1476:∞ 1461:∑ 1431:‖ 1418:‖ 1413:∞ 1398:∑ 1367:) 1363:• 1302:, 1298:, 1294:, 1290:, 1255:) 1162:) 1128:) 1067:) 998:) 864:-- 773:• 646:-- 626:⋅ 617:− 599:− 520:. 476:• 329:| 300:| 291:! 275:| 253:| 180:| 5405:( 5376:( 5357:( 5325:( 5306:( 5278:( 5270:. 5197:( 5164:( 5096:( 5064:( 5031:( 4993:( 4972:. 4964:2 4960:y 4956:, 4953:x 4941:b 4935:+ 4927:1 4923:y 4919:, 4916:x 4904:a 4898:= 4890:2 4886:y 4882:b 4879:+ 4874:1 4870:y 4866:a 4863:, 4860:x 4836:x 4818:0 4809:x 4806:, 4803:x 4786:: 4764:. 4758:y 4755:, 4750:2 4746:x 4739:b 4736:+ 4730:y 4727:, 4722:1 4718:x 4711:a 4708:= 4702:y 4699:, 4694:2 4690:x 4686:b 4683:+ 4678:1 4674:x 4670:a 4653:, 4651:b 4647:a 4614:( 4598:( 4572:( 4554:( 4518:( 4497:( 4488:/ 4484:/ 4462:{ 4436:( 4421:( 4407:( 4392:( 4378:( 4370:x 4366:x 4361:x 4352:) 4344:( 4325:( 4317:) 4309:( 4294:( 4276:) 4268:( 4253:( 4233:( 4218:( 4195:) 4187:( 4171:( 4150:( 4134:( 4089:( 4063:( 4045:( 4024:2 4014:i 4010:x 4001:I 3995:i 3970:( 3944:i 3940:H 3934:I 3928:i 3897:( 3873:( 3842:( 3825:H 3819:H 3813:x 3807:x 3801:x 3795:H 3773:( 3756:( 3717:( 3694:( 3677:( 3630:( 3568:( 3550:( 3506:) 3502:( 3486:( 3448:( 3431:( 3353:( 3337:( 3322:( 3306:( 3274:( 3258:( 3242:( 3190:C 3173:M 3139:" 3135:" 3108:" 3104:" 3090:" 3086:" 3072:" 3051:( 3043:( 3029:( 3016:t 3002:) 2999:t 2996:( 2970:( 2951:C 2934:M 2911:" 2907:" 2895:" 2891:" 2886:? 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