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The section "bases" is reproduced verbatim in the article orthonormal basis. I assume this is done on purpose, but I don't see the reason for it. I do see a danger however: improvements in this article may not be incorperated in the original, and vice versa.
In the same section, I don't understand the sentence "only countably many terms in this sum [referring to the Fourier series] will be non-zero, and the expression is therefore well-defined". I suppose "well-defined" means that the sum is finite, but this does not follow immediately from the fact that the number of non-zero summands is countable. Am I misunderstanding the sentence? -- Jitse Niesen 15:17, 1 Mar 2004 (UTC)
This article seems to assume a great deal more knowledge in the reader than can reasonably be expected. Obviously, when you're dealing with abstract mathematical concepts, it gets difficult to explain things in general terms, and without referencing other abstruse concepts and vocabulary. Still, I think Wikipedia can do a lot better than this (and it has-- see: Quantum Mechanics). After reading the article, I still had almost no idea of what a Hilbert space actually is. I can't imagine anyone who hasn't studied higher mathematics getting any use out of the article in its current form.
Thankfully, I managed to find a satisfactory explanation here. I think this too might be beyond what an average person can follow, but it's a good example of how to explain a mathematical concept in terms of physical phenomonon, and without reverting to mathematicianese. It also gives much more background to the Hilbert space than this article affords it, and which the subject deserves.
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.
It's one thing to say my article is not "good", it's quite another to say I am a "crackpot". I challenge Lethe to give even ONE specific example in which I wrote something about physics that is "crackpot". Making errors as all do is not same as being "crackpot". I use only mainstream physics. I do not say that relativity is wrong. I use relativity. Ditto with quantum theory. So what is Lethe talking about? BTW:
Do you mean "Hamilton formulated a new description of classical mechanics which was eventually housed in an infinite-dimensional phase space. In this space, a point represents the entire physical system."
What's wrong with that? I am not saying phase space is Hilbert space. Is that what you assumed? In any case that statement hardly makes me a "crackpot". The phase space of classical fields is infinite.
Looking at Lethe's page he seems to be a mathematician with the typical arrogance mathematicians have toward theorertical physicists and vice versa. Feynman referred to their penchant for "rigor mortis" that is evident in Lethe's remarks here. I have a PhD in physics from the University of California BTW. Lethe seems to be part of John Baez's clique trying to take over the Internet Math/Physics which explains his "crackpot" remark since John Baez has held a personal grudge against me for more than ten years including spreading a false story about me and Gell-Mann, which never happened. What Baez did do was to garble a story Ed Siegel told me about him and Gell-Mann. Baez simply substituted my name where Ed Siegel's should have been. Jack Sarfatti JackSarfatti 00:12, 28 October 2005 (UTC)!
I've noticed that nearly all articles of a scientific or mathematical nature on wikipedia are nearly indecipherable to the kind of person who would most likely be referencing the subject on wikipedia. While I'm sure the article is well-researched, I doubt that the kind of person who could decipher most of what's in this page would be looking it up on Wikipedia.
Now, that being said, it is true that part of the wikipedia concept is that it should be a storehouse of knowledge. I'm therefore not suggesting that anything in this or other articles of a similar nature should be deleted; as a "record" of knowledge it seems to stand pat (though I'll let the experts dispute that if they want, I have no idea myself, I'm just speaking in terms of content style); but there should be an effort to include a seperate "layperson's" section in these types of articles that explains the basic concepts to the person most likely looking up this type of article. In this case it is most likely somebody who is trying to learn about qunatum physics and has stumbled across the term "hilbert space", has no idea what it means, and goes to wikipedia to unsuccesfully find clarity.
And, no, I'm not volunteering, because I'm a layperson on this subject myself. I've certainly endeavoured to do this in my area of speciality. If somebody wants to test their teaching skills, I'd love for somebody to decrypt these types of things. (Unsigned comment from User:Sayfadeen 17 January 2006).
I got no clue what a hilbert space is by reading the article, i'm not a scientist. I've got some math at school but not this much. Isn't there a simple kind of figure who might explain this 'space' thing since it's called space i asume it's a kind of shape. Perhapps with some strange future's. I'm just like the 95% of the world who believes a picture can tell more then 1000 words. pgt2006 2 feb 2006
Listen you guys, it's hard enough to write a math article that's correct, it's harder still to make it correct, complete, and still as comprehensible to the non-expert as it can be. It's quite a task! While no one wants to make their articles inaccessible, it's certainly incorrect to say that experts do not use Wikipedia. We serve experts and nonexperts alike here. Thus, completeness and correctness are always important goals, and they're probably easier to attain. We serve nonexperts as well, so that accessibility is an important goal as well, though it might be harder. Which goals are more important is the subject of various debate, suffice to say, this is well-worn territory, so you don't have to theorize about the relevance of super-technical material on wikipedia. Let's focus much more specifically: this article assumes too much prior knowledge right from the first sentence, and needs help. We can certainly do that, let's give it a try. Now, I've just rewritten the intro paragraph. It would be helpful if you would comment on the new intro. Also, perhaps you could describe exactly which parts are too technical to your eye. The technical matter can't be forgotten, but it can certainly be postponed and contextualized. My point is, instead of complaining about how inaccessible it is, help us to improve it. Even if you don't feel you're qualified to rewrite it yourselves, you can help just by saying which words are too technical too soon. This is a wiki, afterall! - lethe talk + 21:34, 2 February 2006 (UTC)
I am a physicist and i have started to use wikipedia cause it has grown to a point where it is quite useful. Its a convenient way to remind me of basic things that I may have forgortten, or didnt learn that well in the first place. Thanks to you-all! Physicists use Hilbert spaces and other n-dimensional spaces to make it easier for them to *mathematically* manipulate and describe physical phenomena to other physicists. They are tools of mathematical algebraic convenience and impossible to describe in a 3 dimensional euclidean space picture. As such, they are of no use and simply confusing for lay people. 69.44.253.177 01:46, 14 August 2006 (UTC)
"If a linear operator is defined on all of a Hilbert space then it is necessarily bounded. However, if we allow ourselves to define a linear map that is defined on a proper subspace of the Hilbert space, then we can obtain unbounded operators."
I've deleted this, http://en.wikipedia.org/wiki/Discontinuous_linear_map gives a general example of unbounded operator on any infinite dimensional normed space into any non-zero normed space, so this is not true. —The preceding unsigned comment was added by Scineram ( talk • contribs) .
"Completeness in this context means that every Cauchy sequence of elements of the space converges to an element in the space, in the sense that the norm of differences approaches zero."
This confused me for a while until I realized "norm of differences" meant "difference of the norms" (or at least I think that's what it means). Based on my background knowledge of a Cauchy sequences, I think we mean to say that any sequence of vectors in Hilbert space H that has the property such that the sequence of norms corresponding to those vectors is Cauchy will converge to the norm of a vector in H.
I.e. v1, v2, v3, v4... will have corresponding norms: n1, n2, n3, n4.... Since the norms are scalar numbers, we define them being Cauchy in the usual way, and the sequence of norms: n1, n2, n3, n4... must converge to the norm of some vector in H.
If I have guessed right, I move to change "norm of differences" to "difference of norms" or even "difference of the norm of the vector being approached and the norms of the vectors in the sequence", as the article doesn't mention what two things the difference is of. Loodog 01:15, 7 March 2006 (UTC)
Small Question - Can anyone show me an example of a situation in which "bra-ket notation . . . is frowned upon my mathematicians". While I am in the dept of physics, I've never heard any of the math-folks say a single bad thing about. Granted, they wouldn't use it to do proofs or anything but they have all acknowledged it's amazing utility. Wrath0fb0b 17:07, 9 March 2006 (UTC)
"Since all infinite-dimensional separable Hilbert spaces are isomorphic"
Isomorphic as what? And are separable and inseparable spaces isomorphic? —The preceding unsigned comment was added by 59.144.16.174 ( talk • contribs) .
I am reading Alain Connes' Noncommutative Geometry, and in the first chapter he writes an innocent sounding statement (paraphrased): up to isomorphism, there is only one Hilbert Space with a countable basis.
"Countable basis" is a topological term, and the "basis" used here appears to be the linear algebra context. There does not appear to be a mention of the topological underlying of a Hilbert space on this page. I doubt its insignificant. Should that be added? I would do it, but I came here to find the answer to my question so I'm obviously not the best person. -- 68.98.221.241 22:58, 30 June 2006 (UTC)
Could someone please revise the Definition section of this article in the following ways:
-- DrEricH 23:09, 21 August 2006 (UTC)
Firstly, I would like to mention that I am not a mathematician. The last time I covered Hilbert spaces was in an undergraduate Linear Algebra course over 20 years ago and I only understood the concept in the context of quantum mechanics. Therefore I will not comment on content, but on editorial standards and style. In my opinion, this is a Good Article, however, I would like to make a few suggestions for improvement:
I have placed the article “On hold” for the changes to be implemented. RelHistBuff 10:03, 10 October 2006 (UTC)
The statement that the bra-ket is "frowned upon" by mathematicians seems a little histrionic. Just because something is not used doesn't make it "frowned upon." Moreover, in a couple of places, it was suggested that infinite dimensional Hilbert spaces are the only ones that matter. That is definitely not true: many authors use Hilbert spaces when they could, and often do, mean finite dimensional spaces so that they can treat the finite and infinite dimensional cases together. – Joke 15:03, 10 October 2006 (UTC)
I wanted to suggest maybe opening the section on the definition slightly differently. Something like "A Hilbert space is a vector space with some additional stucture. In particular it has an inner product and it is complete with respect to topology created by this inner product. To understand what this means notice that every inner product....." I am not sure what I have written is any good but I feel it could use a bit of a description of where we are going before we start speaking of open balls and topology. Any thoughts Thenub314 12:44, 11 October 2006 (UTC)
I uploaded the image File:Completeness in Hilbert space.png a few minutes ago. When I realized that the yellow arrow is nearly invisible in the thumbnail rendering, I edited the image to thicken the lines. The thumbnail doesn't seem to have updated, though. Does anyone know how to force the server to generate a new thumbnail? "Purging the cache" sounds promising, but a naive attempt to do this has no apparent effect. Sławomir Biały ( talk) 15:52, 11 September 2009 (UTC)
[Content transferred from Talk:Hilbert space/GA2.]
"Relative to a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a pre-Hilbert space." in the Definition section should be rewritten.
the sentence underneath the picture with the broken line are not correct, i think.
"Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in yellow)."
The idee behind this sentence is not equivalent with completeness, and so raises confussion, or does somebody have a proof of this?...
An maybe its a good idee to clean up the discusion page. greetings S. —Preceding unsigned comment added by 157.193.53.246 ( talk • contribs)
I reverted the edit that moved the example down past the definition. This example is likely to be something that all readers will be able to grasp, and so should be before the formal definition. This is the opposite order that one is used to seeing things in mathematics textbooks (which Wikipedia is not), but I do think that it is more likely to be meaningful to a larger number of prospective readers. The model for the article was based on Group (mathematics), a mathematics featured article, which also gives a toy example to clarify the concept before the definition. Sławomir Biały ( talk) 23:08, 26 May 2010 (UTC)
Sławomir Biały: I'm happy you included Folland's book as a reference. This work is a favorite of mine for a clear exposition of the subject. Brews ohare ( talk) 17:59, 24 June 2010 (UTC)
In a pair of edits ( here and again here), the term Euclidean space was removed in favor of other possibilities: "finite dimensional Euclidean space" and "n-space". However, I think that Euclidean space should be restored. For a totally non-mathematical audience, this conveys precisely that Hilbert spaces are spaces in which one can still perform Euclidean geometry, whereas "n-space" conveys nothing meaningful whatsoever to such an audience, and "finite dimensional Euclidean space" seems overly complicated. I realize that there are mathematicians that also consider infinite-dimensional Euclidean spaces (in fact, these are what we call "Hilbert spaces" here), but the finite dimensional case is still the primary use of the term (both within mathematics, and certainly outside it). Anyway, we needn't be overly concerned for those who "know better": they will certainly be able to cope as well. I think that perhaps these edits have missed the point that the lead section of an article should be a general-audience description and not one that is intended to be a mathematically precise characterization. Sławomir Biały ( talk) 14:26, 5 October 2010 (UTC)
In the "Notes-Section" at point 46 is referred to: Dunford & Schwartz 1958, II.4.29 Well, I looked for that reference, but I cannot find it. Here is the book at Amazon: [1] There is simply no section "4", but there are many... Have I picked the wrong book or is the reference incorrect? -- Vilietha ( talk) 14:44, 25 April 2011 (UTC)
I'm impressed that I clicked on the wikilink of a maths term I didn't understand, read the first paragraph of this article, and now understand enough to go back and finish the article I was reading. This is how wikipedia should be! Thanks guys. -- Physics is all gnomes ( talk) 00:09, 28 December 2010 (UTC)
I would just like to say thanks to this talk page I finally get Hilbert Spaces, much easier to glean concepts from a few arguments than the article itself. I'm a PhD Student Cheers :) -- 78.86.197.227 ( talk) 17:06, 4 May 2011 (UTC)
"...to spaces with any number of dimensions (or coordinate axes), including potentially infinitely many dimensions..." What is meant by "potentially" here? Just "possibly", or indeed something in the spirit of Actual infinity#Aristotle's Potential-Actual Distinction? Boris Tsirelson ( talk) 17:48, 9 September 2009 (UTC)
The author references Kolmogorov-Fomin Real Analysis, Prentice Hall (1970.) It may be confusing to some readers who know the book, that Kolmogorov defines Hilbert Spaces to be strictly infinite dimensional. On p. 155 of K&F Hilbert Space is defined as "A Euclidean Space which is complete, separable, and infinite-dimensional." Other classic references also use the Kolmogorov definition. e.g. Dennery and Krzywicki Mathematics for Physicists Dover (1995) p.197 (seems to). In many (older) references, finite dimensional Hilbert Spaces are not defined. Probably useful to readers to acknowledge the various historical usages of the term, especially so, for an encyclopedia article. A few words contrasting the various definitions would seem appropriate. BTW, nice work on this detailed article.
suggestion: "Some older references define Hilbert Spaces to be strictly infinite dimensional. e.g. Kolmogorov and Fomin (1970)." — Preceding unsigned comment added by Mathview2011 ( talk • contribs) 00:58, 27 May 2011 (UTC)
There are three kinds of such brackets in Unicode, see Bracket (mathematics). The correct characters for mathematical angle brackets are ⟨mathematical left/right angle brackets⟩ (U+27e8 and U+27e9). The HTML entities lang and rang resolve to left/right-pointing angle bracket (U+2329 and U+232a), which are deprecated by Unicode because they are canonical equivalent to Chinese punctuation (U+3008 and U+3009). For a proof of why we must not use lang and rang entities, see the revision 2011-08-22T05:57:55 by Headbomb. I doubt he replaced them intentionally, so I conclude that some Unicode aware software may replace them automatically with the characters inappropriate for mathematical text. The root of the problem is that lang and rang don't have the correct semantic information.
Hereby, in order to avoid any further debate, I replace all angle brackets with LaTeX as required by convention. bungalo ( talk) 08:56, 30 August 2011 (UTC)
Previous discussion: Talk:Hilbert_space/Archive_2#Angle_brackets -- LutzL ( talk) 11:00, 30 August 2011 (UTC)
Can we please avoid incorrectly suggesting that every hilbert space is a vector space? Vector spaces are generated by finite combinations of bases, while the finite restriction is lifted in hilbert spaces. I know it is a technical detail, and colloquial verbal usage frequently ignores the difference -- but let's be correct. -- Liuyipei ( talk) 08:43, 29 February 2012 (UTC)
At Goettingen, the math building's foyer, where students normally first enter, is called "Hilbertraum" [Hilbert room/space]. According to the German Wik, other German universities also have Hilbertraeume. should this be incorporated into our article? Kdammers ( talk) 03:38, 29 April 2012 (UTC)
"the state space for position and momentum states is..." — seems ugly; someone could think that a "position state" is the state localized at a point (but in fact, there is no such state in this space). Maybe , rather something like "the state space for a single nonrelativistic spinless particle moving in the three-dimensional Euclidean space"?
"while the state space for the spin of a single proton is just the product of two complex planes." — or maybe "while the state space for the spin of a single spin-1/2 particle (for example, proton) is two-dimensional (just the product of two complex planes)."
"Each observable is represented by a maximally-Hermitian (precisely: by a self-adjoint) linear operator" — I do not remember the term "maximally-Hermitian"; what is it? Also, "Hermitian" usually assumes "bounded", while "self-adjoint" does not.
"If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues." — Yes; and if it is not discrete then what? Then the observable can only attain values belonging to the spectrum.
"During a measurement, the probability that a system collapses..." — Yes, but all that is about an ideal quantum measurement (which is much simpler that the general case described via "ancillas", "effects", "quantum operations", "quantum instruments" etc).
"Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute." — Really? Does it also (or rather, mainly) treats some, quite nontrivial implications of the non-commuting? And not quite from non-commuting, but rather, from the commutant being a scalar operator.
Boris Tsirelson ( talk) 12:29, 11 September 2009 (UTC)
This section systematically refers to the Hilbert space as the "state space" of quantum mechanics. Formally speaking, this is wrong. Formally, the space of states is the space of positive linear functionals on the operator algebra that send the unit operator to 1. This space is generally much larger than just the unit elements of the Hilbert space (which it naturally contains). The states that are related to Hilbert space elements are called "pure". All other states can be written as linear combinations of pure states and are called "mixed". The proper classical analogue of the Hilbert space is phase space, with the operators corresponding to functions on this phase space and state being distributions on the phase space. Now, this is mostly formally and it is not unusual for physicists to refer to elements of the Hilbert space as states, I would just hold of on calling the Hilbert space "the state space". ( TimothyRias ( talk) 08:53, 23 September 2009 (UTC))
A mathematician I used to work with, Kane Yee (best known for a numerical algorithm), said that the space of electromagnetic waves is actually only a Banach space because the evanescent and free solutions have different normalization. This appears to apply also to qm. Bound states of negative energy and free states of positive energy are normalized differently. This is "fixed" by introducing a fictitious box, making all states bound. Julian Swinger said that one actually uses four such boxes, each infinite with respect to what is inside it, but the lecture series was cancelled before I learned what the other three are. David R. Ingham ( talk) 04:45, 10 June 2012 (UTC)
Hilbert dimension is an algebraic concept, from ring theory. Hilbert space dimension seems to be more common for the dimension of a Hilbert space in my opinion. -- Chricho ∀ ( talk) 16:05, 3 August 2012 (UTC)
“ | "(...) The
Occam's razor is a principle of parsimony, economy, or succinctness used in logic and problem-solving. It states that among competing hypotheses, the hypothesis with the fewest assumptions should be selected. (...)" |
” |
— " Hilbert space" article |
Hallo there Prof.
B. Tsirelson,
I hope my explanation helps.
Please let's try to have a nice and relaxed weekend.
Cheers.
M aurice
Carbonaro 09:53, 19 April 2013 (UTC)
Talking about the representation of elements within a Hilbert space, the introduction says:
In fact this would also hold for any orthogonal basis, and even for any vector set that forms a basis, without the need for them to be either orthogonal nor normal. Besides this, I feel the analogy with the Cartesian plane is a little bit low-level, and I think it would be more appropriate in the article explaining what a base is.
I would either remove the entire sentence, for it applies to any vector space and this is not a particular property of Hilbert spaces alone, or would at least remove the word "orthonormal".
Using the same reasoning, I would like to reconsider this affirmation:
If the chosen base is not orthogonal, could we say that linear operators stretch the space in mutually perpendicular directions?
Elferdo ( talk) 15:35, 22 July 2013 (UTC)
BTW, the entire passage in the lead about linear operators looks very dubious to me. Over complex numbers a linear operator is not necessarily self-adjoint, whereas over real numbers it may have an empty spectrum, but even if sufficient number of eigenvectors exist to form a basis, they are not necessarily orthogonal. I wonder how this crap managed to find its way into this so named “good” article in spite of all this crowd of experts here. Incnis Mrsi ( talk) 14:47, 23 July 2013 (UTC)
This article never mentions the no-cloning theorem or its cousins.It seems to me that this is an important property that *all* (finite-dimensional) Hilbert stpaces have. Viz, you cannot clone some abitrary vector in a Hilbert space, you can only clone the basis vectors (which in turn implies that basis vector must be orthogonal) ... the no-deleting theorem implies that the basis vectors form a complete set ... I mention this because 1) in dagger compact category, these two theorems are used to define the basis, and 2) the dagger compact cat is complete in Hilbert spaces: any theorem that holds for hilbert spaces holds for any dagger compact category, in general ... there should be some blurble of this in this article, but I'm not feeling "bold", as they say. (this article already being quite extensive and thorough -- good job!)
Hmm. I guess a point of confusion would be "what does it mean to clone", as a college student would say "easy, whip out pen and paper, and write it down twice". So somehow, need to get across the idea that if one has two identical hilbert spaces, and has a vector in one, there is no way to take a general vector in one and copy it into the other (short of specifying an infinite number of decimal places in some basis ... Hmmm ... I don't see a good, simple, pedestrian explanation at the moment...) User:Linas ( talk) 19:05, 27 November 2013 (UTC)
"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. Dratman ( talk) 02:21, 2 April 2014 (UTC)
In the article, the general expression is given as
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 - A course in Functional Analysis. Should this be mentioned? (vn article) YohanN7 ( talk) 14:03, 12 April 2014 (UTC)
How do complex affine spaces relate to complex Hilbert spaces? — Cheers, Steelpillow ( Talk) 12:32, 30 September 2015 (UTC)
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.
Maybe I'll comeback for a more detailed pass later. ( TimothyRias ( talk) 08:26, 8 September 2009 (UTC))
I have added a paragraph with a reference to the Separable spaces section to address your second comment. Sławomir Biały ( talk) 12:19, 8 September 2009 (UTC)
As for me, your remarks about nonseparability in QFT do not conform to the fact that the Fock space is separable (and your arguments do not involve non-Fock spaces). Boris Tsirelson ( talk) 13:58, 8 September 2009 (UTC)
Some early feedback, some acronyms are introduced out of the blue (PDE for partial differential equation, L2 is dropped without proper introduction, and so on). Now I know what these are, but I also have a background in physics. Also I notice nothing is mentioned about Banach spaces and recall hearing that Hilbert & Banach spaces were intimately related (never worked with Banach spaces, so I could very well be 20 miles off track here!). Headbomb { ταλκ κοντριβς – WP Physics} 17:36, 8 September 2009 (UTC)
great input - if I was in a graduate class in topology this article works, but MOST are not — Preceding unsigned comment added by 75.163.175.56 ( talk) 18:16, 27 December 2015 (UTC)
See Stefan Bergman and Bergman kernel, not Valentine Bargmann and Segal–Bargmann space. Boris Tsirelson ( talk) 18:37, 10 May 2016 (UTC)
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 ? — Preceding unsigned comment added by 64.180.21.136 ( talk) 01:50, 29 October 2016 (UTC)
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.
It is not a exceptional case; much mathematics is developed with that other (hidden) convention.
It follows from properties 1 and 2 that a complex inner product is antilinear in its second argument, meaning that
Álvaro López de Quadros ( talk) 20:59, 1 March 2017 (UTC) Footnotes
The lack of precision in many mathematical texts of Wikipedia really is astonishing.
"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.
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 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. — Preceding unsigned comment added by 217.95.164.135 ( talk) 21:20, 29 September 2019 (UTC)
The Clarify-jargon template I added was summarily removed as a "drive-by tagging", whatever that is. So without re-adding it, let me explain why I put it there in the first place: I have no idea what Hilbert space is and I have tried to get through this article several times. Since I am a smart person who studied maths to A-level I conclude that my experience is not unrepresentative. To understand the first paragraph, you are asking readers to get a grip on: Euclidean space, vector space, complete metric space, mathematical series, absolute convergence, and norm (mathematics). It's just too much. My feeling is that you need something much more general:
...with further technical details coming in gradually lower down. The whole article has similar problems, however, which is why I tagged it. Widsith ( talk) 15:54, 8 September 2009 (UTC)
I have rewritten the first paragraph to address this concern. Sławomir Biały ( talk) 18:18, 8 September 2009 (UTC)
I've just discovered my book uses (f,g)=equation and <f,g>=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).
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. — Preceding unsigned comment added by 2601:143:480:a4c0:6dd5:ec40:6d14:9d00 ( talk) 10:04, 2 April 2020 (UTC)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 |
The section "bases" is reproduced verbatim in the article orthonormal basis. I assume this is done on purpose, but I don't see the reason for it. I do see a danger however: improvements in this article may not be incorperated in the original, and vice versa.
In the same section, I don't understand the sentence "only countably many terms in this sum [referring to the Fourier series] will be non-zero, and the expression is therefore well-defined". I suppose "well-defined" means that the sum is finite, but this does not follow immediately from the fact that the number of non-zero summands is countable. Am I misunderstanding the sentence? -- Jitse Niesen 15:17, 1 Mar 2004 (UTC)
This article seems to assume a great deal more knowledge in the reader than can reasonably be expected. Obviously, when you're dealing with abstract mathematical concepts, it gets difficult to explain things in general terms, and without referencing other abstruse concepts and vocabulary. Still, I think Wikipedia can do a lot better than this (and it has-- see: Quantum Mechanics). After reading the article, I still had almost no idea of what a Hilbert space actually is. I can't imagine anyone who hasn't studied higher mathematics getting any use out of the article in its current form.
Thankfully, I managed to find a satisfactory explanation here. I think this too might be beyond what an average person can follow, but it's a good example of how to explain a mathematical concept in terms of physical phenomonon, and without reverting to mathematicianese. It also gives much more background to the Hilbert space than this article affords it, and which the subject deserves.
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.
It's one thing to say my article is not "good", it's quite another to say I am a "crackpot". I challenge Lethe to give even ONE specific example in which I wrote something about physics that is "crackpot". Making errors as all do is not same as being "crackpot". I use only mainstream physics. I do not say that relativity is wrong. I use relativity. Ditto with quantum theory. So what is Lethe talking about? BTW:
Do you mean "Hamilton formulated a new description of classical mechanics which was eventually housed in an infinite-dimensional phase space. In this space, a point represents the entire physical system."
What's wrong with that? I am not saying phase space is Hilbert space. Is that what you assumed? In any case that statement hardly makes me a "crackpot". The phase space of classical fields is infinite.
Looking at Lethe's page he seems to be a mathematician with the typical arrogance mathematicians have toward theorertical physicists and vice versa. Feynman referred to their penchant for "rigor mortis" that is evident in Lethe's remarks here. I have a PhD in physics from the University of California BTW. Lethe seems to be part of John Baez's clique trying to take over the Internet Math/Physics which explains his "crackpot" remark since John Baez has held a personal grudge against me for more than ten years including spreading a false story about me and Gell-Mann, which never happened. What Baez did do was to garble a story Ed Siegel told me about him and Gell-Mann. Baez simply substituted my name where Ed Siegel's should have been. Jack Sarfatti JackSarfatti 00:12, 28 October 2005 (UTC)!
I've noticed that nearly all articles of a scientific or mathematical nature on wikipedia are nearly indecipherable to the kind of person who would most likely be referencing the subject on wikipedia. While I'm sure the article is well-researched, I doubt that the kind of person who could decipher most of what's in this page would be looking it up on Wikipedia.
Now, that being said, it is true that part of the wikipedia concept is that it should be a storehouse of knowledge. I'm therefore not suggesting that anything in this or other articles of a similar nature should be deleted; as a "record" of knowledge it seems to stand pat (though I'll let the experts dispute that if they want, I have no idea myself, I'm just speaking in terms of content style); but there should be an effort to include a seperate "layperson's" section in these types of articles that explains the basic concepts to the person most likely looking up this type of article. In this case it is most likely somebody who is trying to learn about qunatum physics and has stumbled across the term "hilbert space", has no idea what it means, and goes to wikipedia to unsuccesfully find clarity.
And, no, I'm not volunteering, because I'm a layperson on this subject myself. I've certainly endeavoured to do this in my area of speciality. If somebody wants to test their teaching skills, I'd love for somebody to decrypt these types of things. (Unsigned comment from User:Sayfadeen 17 January 2006).
I got no clue what a hilbert space is by reading the article, i'm not a scientist. I've got some math at school but not this much. Isn't there a simple kind of figure who might explain this 'space' thing since it's called space i asume it's a kind of shape. Perhapps with some strange future's. I'm just like the 95% of the world who believes a picture can tell more then 1000 words. pgt2006 2 feb 2006
Listen you guys, it's hard enough to write a math article that's correct, it's harder still to make it correct, complete, and still as comprehensible to the non-expert as it can be. It's quite a task! While no one wants to make their articles inaccessible, it's certainly incorrect to say that experts do not use Wikipedia. We serve experts and nonexperts alike here. Thus, completeness and correctness are always important goals, and they're probably easier to attain. We serve nonexperts as well, so that accessibility is an important goal as well, though it might be harder. Which goals are more important is the subject of various debate, suffice to say, this is well-worn territory, so you don't have to theorize about the relevance of super-technical material on wikipedia. Let's focus much more specifically: this article assumes too much prior knowledge right from the first sentence, and needs help. We can certainly do that, let's give it a try. Now, I've just rewritten the intro paragraph. It would be helpful if you would comment on the new intro. Also, perhaps you could describe exactly which parts are too technical to your eye. The technical matter can't be forgotten, but it can certainly be postponed and contextualized. My point is, instead of complaining about how inaccessible it is, help us to improve it. Even if you don't feel you're qualified to rewrite it yourselves, you can help just by saying which words are too technical too soon. This is a wiki, afterall! - lethe talk + 21:34, 2 February 2006 (UTC)
I am a physicist and i have started to use wikipedia cause it has grown to a point where it is quite useful. Its a convenient way to remind me of basic things that I may have forgortten, or didnt learn that well in the first place. Thanks to you-all! Physicists use Hilbert spaces and other n-dimensional spaces to make it easier for them to *mathematically* manipulate and describe physical phenomena to other physicists. They are tools of mathematical algebraic convenience and impossible to describe in a 3 dimensional euclidean space picture. As such, they are of no use and simply confusing for lay people. 69.44.253.177 01:46, 14 August 2006 (UTC)
"If a linear operator is defined on all of a Hilbert space then it is necessarily bounded. However, if we allow ourselves to define a linear map that is defined on a proper subspace of the Hilbert space, then we can obtain unbounded operators."
I've deleted this, http://en.wikipedia.org/wiki/Discontinuous_linear_map gives a general example of unbounded operator on any infinite dimensional normed space into any non-zero normed space, so this is not true. —The preceding unsigned comment was added by Scineram ( talk • contribs) .
"Completeness in this context means that every Cauchy sequence of elements of the space converges to an element in the space, in the sense that the norm of differences approaches zero."
This confused me for a while until I realized "norm of differences" meant "difference of the norms" (or at least I think that's what it means). Based on my background knowledge of a Cauchy sequences, I think we mean to say that any sequence of vectors in Hilbert space H that has the property such that the sequence of norms corresponding to those vectors is Cauchy will converge to the norm of a vector in H.
I.e. v1, v2, v3, v4... will have corresponding norms: n1, n2, n3, n4.... Since the norms are scalar numbers, we define them being Cauchy in the usual way, and the sequence of norms: n1, n2, n3, n4... must converge to the norm of some vector in H.
If I have guessed right, I move to change "norm of differences" to "difference of norms" or even "difference of the norm of the vector being approached and the norms of the vectors in the sequence", as the article doesn't mention what two things the difference is of. Loodog 01:15, 7 March 2006 (UTC)
Small Question - Can anyone show me an example of a situation in which "bra-ket notation . . . is frowned upon my mathematicians". While I am in the dept of physics, I've never heard any of the math-folks say a single bad thing about. Granted, they wouldn't use it to do proofs or anything but they have all acknowledged it's amazing utility. Wrath0fb0b 17:07, 9 March 2006 (UTC)
"Since all infinite-dimensional separable Hilbert spaces are isomorphic"
Isomorphic as what? And are separable and inseparable spaces isomorphic? —The preceding unsigned comment was added by 59.144.16.174 ( talk • contribs) .
I am reading Alain Connes' Noncommutative Geometry, and in the first chapter he writes an innocent sounding statement (paraphrased): up to isomorphism, there is only one Hilbert Space with a countable basis.
"Countable basis" is a topological term, and the "basis" used here appears to be the linear algebra context. There does not appear to be a mention of the topological underlying of a Hilbert space on this page. I doubt its insignificant. Should that be added? I would do it, but I came here to find the answer to my question so I'm obviously not the best person. -- 68.98.221.241 22:58, 30 June 2006 (UTC)
Could someone please revise the Definition section of this article in the following ways:
-- DrEricH 23:09, 21 August 2006 (UTC)
Firstly, I would like to mention that I am not a mathematician. The last time I covered Hilbert spaces was in an undergraduate Linear Algebra course over 20 years ago and I only understood the concept in the context of quantum mechanics. Therefore I will not comment on content, but on editorial standards and style. In my opinion, this is a Good Article, however, I would like to make a few suggestions for improvement:
I have placed the article “On hold” for the changes to be implemented. RelHistBuff 10:03, 10 October 2006 (UTC)
The statement that the bra-ket is "frowned upon" by mathematicians seems a little histrionic. Just because something is not used doesn't make it "frowned upon." Moreover, in a couple of places, it was suggested that infinite dimensional Hilbert spaces are the only ones that matter. That is definitely not true: many authors use Hilbert spaces when they could, and often do, mean finite dimensional spaces so that they can treat the finite and infinite dimensional cases together. – Joke 15:03, 10 October 2006 (UTC)
I wanted to suggest maybe opening the section on the definition slightly differently. Something like "A Hilbert space is a vector space with some additional stucture. In particular it has an inner product and it is complete with respect to topology created by this inner product. To understand what this means notice that every inner product....." I am not sure what I have written is any good but I feel it could use a bit of a description of where we are going before we start speaking of open balls and topology. Any thoughts Thenub314 12:44, 11 October 2006 (UTC)
I uploaded the image File:Completeness in Hilbert space.png a few minutes ago. When I realized that the yellow arrow is nearly invisible in the thumbnail rendering, I edited the image to thicken the lines. The thumbnail doesn't seem to have updated, though. Does anyone know how to force the server to generate a new thumbnail? "Purging the cache" sounds promising, but a naive attempt to do this has no apparent effect. Sławomir Biały ( talk) 15:52, 11 September 2009 (UTC)
[Content transferred from Talk:Hilbert space/GA2.]
"Relative to a distance function defined in this way, any inner product space is a metric space, and sometimes is known as a pre-Hilbert space." in the Definition section should be rewritten.
the sentence underneath the picture with the broken line are not correct, i think.
"Completeness means that if a particle moves along the broken path (in blue) travelling a finite total distance, then the particle has a well-defined net displacement (in yellow)."
The idee behind this sentence is not equivalent with completeness, and so raises confussion, or does somebody have a proof of this?...
An maybe its a good idee to clean up the discusion page. greetings S. —Preceding unsigned comment added by 157.193.53.246 ( talk • contribs)
I reverted the edit that moved the example down past the definition. This example is likely to be something that all readers will be able to grasp, and so should be before the formal definition. This is the opposite order that one is used to seeing things in mathematics textbooks (which Wikipedia is not), but I do think that it is more likely to be meaningful to a larger number of prospective readers. The model for the article was based on Group (mathematics), a mathematics featured article, which also gives a toy example to clarify the concept before the definition. Sławomir Biały ( talk) 23:08, 26 May 2010 (UTC)
Sławomir Biały: I'm happy you included Folland's book as a reference. This work is a favorite of mine for a clear exposition of the subject. Brews ohare ( talk) 17:59, 24 June 2010 (UTC)
In a pair of edits ( here and again here), the term Euclidean space was removed in favor of other possibilities: "finite dimensional Euclidean space" and "n-space". However, I think that Euclidean space should be restored. For a totally non-mathematical audience, this conveys precisely that Hilbert spaces are spaces in which one can still perform Euclidean geometry, whereas "n-space" conveys nothing meaningful whatsoever to such an audience, and "finite dimensional Euclidean space" seems overly complicated. I realize that there are mathematicians that also consider infinite-dimensional Euclidean spaces (in fact, these are what we call "Hilbert spaces" here), but the finite dimensional case is still the primary use of the term (both within mathematics, and certainly outside it). Anyway, we needn't be overly concerned for those who "know better": they will certainly be able to cope as well. I think that perhaps these edits have missed the point that the lead section of an article should be a general-audience description and not one that is intended to be a mathematically precise characterization. Sławomir Biały ( talk) 14:26, 5 October 2010 (UTC)
In the "Notes-Section" at point 46 is referred to: Dunford & Schwartz 1958, II.4.29 Well, I looked for that reference, but I cannot find it. Here is the book at Amazon: [1] There is simply no section "4", but there are many... Have I picked the wrong book or is the reference incorrect? -- Vilietha ( talk) 14:44, 25 April 2011 (UTC)
I'm impressed that I clicked on the wikilink of a maths term I didn't understand, read the first paragraph of this article, and now understand enough to go back and finish the article I was reading. This is how wikipedia should be! Thanks guys. -- Physics is all gnomes ( talk) 00:09, 28 December 2010 (UTC)
I would just like to say thanks to this talk page I finally get Hilbert Spaces, much easier to glean concepts from a few arguments than the article itself. I'm a PhD Student Cheers :) -- 78.86.197.227 ( talk) 17:06, 4 May 2011 (UTC)
"...to spaces with any number of dimensions (or coordinate axes), including potentially infinitely many dimensions..." What is meant by "potentially" here? Just "possibly", or indeed something in the spirit of Actual infinity#Aristotle's Potential-Actual Distinction? Boris Tsirelson ( talk) 17:48, 9 September 2009 (UTC)
The author references Kolmogorov-Fomin Real Analysis, Prentice Hall (1970.) It may be confusing to some readers who know the book, that Kolmogorov defines Hilbert Spaces to be strictly infinite dimensional. On p. 155 of K&F Hilbert Space is defined as "A Euclidean Space which is complete, separable, and infinite-dimensional." Other classic references also use the Kolmogorov definition. e.g. Dennery and Krzywicki Mathematics for Physicists Dover (1995) p.197 (seems to). In many (older) references, finite dimensional Hilbert Spaces are not defined. Probably useful to readers to acknowledge the various historical usages of the term, especially so, for an encyclopedia article. A few words contrasting the various definitions would seem appropriate. BTW, nice work on this detailed article.
suggestion: "Some older references define Hilbert Spaces to be strictly infinite dimensional. e.g. Kolmogorov and Fomin (1970)." — Preceding unsigned comment added by Mathview2011 ( talk • contribs) 00:58, 27 May 2011 (UTC)
There are three kinds of such brackets in Unicode, see Bracket (mathematics). The correct characters for mathematical angle brackets are ⟨mathematical left/right angle brackets⟩ (U+27e8 and U+27e9). The HTML entities lang and rang resolve to left/right-pointing angle bracket (U+2329 and U+232a), which are deprecated by Unicode because they are canonical equivalent to Chinese punctuation (U+3008 and U+3009). For a proof of why we must not use lang and rang entities, see the revision 2011-08-22T05:57:55 by Headbomb. I doubt he replaced them intentionally, so I conclude that some Unicode aware software may replace them automatically with the characters inappropriate for mathematical text. The root of the problem is that lang and rang don't have the correct semantic information.
Hereby, in order to avoid any further debate, I replace all angle brackets with LaTeX as required by convention. bungalo ( talk) 08:56, 30 August 2011 (UTC)
Previous discussion: Talk:Hilbert_space/Archive_2#Angle_brackets -- LutzL ( talk) 11:00, 30 August 2011 (UTC)
Can we please avoid incorrectly suggesting that every hilbert space is a vector space? Vector spaces are generated by finite combinations of bases, while the finite restriction is lifted in hilbert spaces. I know it is a technical detail, and colloquial verbal usage frequently ignores the difference -- but let's be correct. -- Liuyipei ( talk) 08:43, 29 February 2012 (UTC)
At Goettingen, the math building's foyer, where students normally first enter, is called "Hilbertraum" [Hilbert room/space]. According to the German Wik, other German universities also have Hilbertraeume. should this be incorporated into our article? Kdammers ( talk) 03:38, 29 April 2012 (UTC)
"the state space for position and momentum states is..." — seems ugly; someone could think that a "position state" is the state localized at a point (but in fact, there is no such state in this space). Maybe , rather something like "the state space for a single nonrelativistic spinless particle moving in the three-dimensional Euclidean space"?
"while the state space for the spin of a single proton is just the product of two complex planes." — or maybe "while the state space for the spin of a single spin-1/2 particle (for example, proton) is two-dimensional (just the product of two complex planes)."
"Each observable is represented by a maximally-Hermitian (precisely: by a self-adjoint) linear operator" — I do not remember the term "maximally-Hermitian"; what is it? Also, "Hermitian" usually assumes "bounded", while "self-adjoint" does not.
"If the operator's spectrum is discrete, the observable can only attain those discrete eigenvalues." — Yes; and if it is not discrete then what? Then the observable can only attain values belonging to the spectrum.
"During a measurement, the probability that a system collapses..." — Yes, but all that is about an ideal quantum measurement (which is much simpler that the general case described via "ancillas", "effects", "quantum operations", "quantum instruments" etc).
"Heisenberg's uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute." — Really? Does it also (or rather, mainly) treats some, quite nontrivial implications of the non-commuting? And not quite from non-commuting, but rather, from the commutant being a scalar operator.
Boris Tsirelson ( talk) 12:29, 11 September 2009 (UTC)
This section systematically refers to the Hilbert space as the "state space" of quantum mechanics. Formally speaking, this is wrong. Formally, the space of states is the space of positive linear functionals on the operator algebra that send the unit operator to 1. This space is generally much larger than just the unit elements of the Hilbert space (which it naturally contains). The states that are related to Hilbert space elements are called "pure". All other states can be written as linear combinations of pure states and are called "mixed". The proper classical analogue of the Hilbert space is phase space, with the operators corresponding to functions on this phase space and state being distributions on the phase space. Now, this is mostly formally and it is not unusual for physicists to refer to elements of the Hilbert space as states, I would just hold of on calling the Hilbert space "the state space". ( TimothyRias ( talk) 08:53, 23 September 2009 (UTC))
A mathematician I used to work with, Kane Yee (best known for a numerical algorithm), said that the space of electromagnetic waves is actually only a Banach space because the evanescent and free solutions have different normalization. This appears to apply also to qm. Bound states of negative energy and free states of positive energy are normalized differently. This is "fixed" by introducing a fictitious box, making all states bound. Julian Swinger said that one actually uses four such boxes, each infinite with respect to what is inside it, but the lecture series was cancelled before I learned what the other three are. David R. Ingham ( talk) 04:45, 10 June 2012 (UTC)
Hilbert dimension is an algebraic concept, from ring theory. Hilbert space dimension seems to be more common for the dimension of a Hilbert space in my opinion. -- Chricho ∀ ( talk) 16:05, 3 August 2012 (UTC)
“ | "(...) The
Occam's razor is a principle of parsimony, economy, or succinctness used in logic and problem-solving. It states that among competing hypotheses, the hypothesis with the fewest assumptions should be selected. (...)" |
” |
— " Hilbert space" article |
Hallo there Prof.
B. Tsirelson,
I hope my explanation helps.
Please let's try to have a nice and relaxed weekend.
Cheers.
M aurice
Carbonaro 09:53, 19 April 2013 (UTC)
Talking about the representation of elements within a Hilbert space, the introduction says:
In fact this would also hold for any orthogonal basis, and even for any vector set that forms a basis, without the need for them to be either orthogonal nor normal. Besides this, I feel the analogy with the Cartesian plane is a little bit low-level, and I think it would be more appropriate in the article explaining what a base is.
I would either remove the entire sentence, for it applies to any vector space and this is not a particular property of Hilbert spaces alone, or would at least remove the word "orthonormal".
Using the same reasoning, I would like to reconsider this affirmation:
If the chosen base is not orthogonal, could we say that linear operators stretch the space in mutually perpendicular directions?
Elferdo ( talk) 15:35, 22 July 2013 (UTC)
BTW, the entire passage in the lead about linear operators looks very dubious to me. Over complex numbers a linear operator is not necessarily self-adjoint, whereas over real numbers it may have an empty spectrum, but even if sufficient number of eigenvectors exist to form a basis, they are not necessarily orthogonal. I wonder how this crap managed to find its way into this so named “good” article in spite of all this crowd of experts here. Incnis Mrsi ( talk) 14:47, 23 July 2013 (UTC)
This article never mentions the no-cloning theorem or its cousins.It seems to me that this is an important property that *all* (finite-dimensional) Hilbert stpaces have. Viz, you cannot clone some abitrary vector in a Hilbert space, you can only clone the basis vectors (which in turn implies that basis vector must be orthogonal) ... the no-deleting theorem implies that the basis vectors form a complete set ... I mention this because 1) in dagger compact category, these two theorems are used to define the basis, and 2) the dagger compact cat is complete in Hilbert spaces: any theorem that holds for hilbert spaces holds for any dagger compact category, in general ... there should be some blurble of this in this article, but I'm not feeling "bold", as they say. (this article already being quite extensive and thorough -- good job!)
Hmm. I guess a point of confusion would be "what does it mean to clone", as a college student would say "easy, whip out pen and paper, and write it down twice". So somehow, need to get across the idea that if one has two identical hilbert spaces, and has a vector in one, there is no way to take a general vector in one and copy it into the other (short of specifying an infinite number of decimal places in some basis ... Hmmm ... I don't see a good, simple, pedestrian explanation at the moment...) User:Linas ( talk) 19:05, 27 November 2013 (UTC)
"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. Dratman ( talk) 02:21, 2 April 2014 (UTC)
In the article, the general expression is given as
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 - A course in Functional Analysis. Should this be mentioned? (vn article) YohanN7 ( talk) 14:03, 12 April 2014 (UTC)
How do complex affine spaces relate to complex Hilbert spaces? — Cheers, Steelpillow ( Talk) 12:32, 30 September 2015 (UTC)
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.
Maybe I'll comeback for a more detailed pass later. ( TimothyRias ( talk) 08:26, 8 September 2009 (UTC))
I have added a paragraph with a reference to the Separable spaces section to address your second comment. Sławomir Biały ( talk) 12:19, 8 September 2009 (UTC)
As for me, your remarks about nonseparability in QFT do not conform to the fact that the Fock space is separable (and your arguments do not involve non-Fock spaces). Boris Tsirelson ( talk) 13:58, 8 September 2009 (UTC)
Some early feedback, some acronyms are introduced out of the blue (PDE for partial differential equation, L2 is dropped without proper introduction, and so on). Now I know what these are, but I also have a background in physics. Also I notice nothing is mentioned about Banach spaces and recall hearing that Hilbert & Banach spaces were intimately related (never worked with Banach spaces, so I could very well be 20 miles off track here!). Headbomb { ταλκ κοντριβς – WP Physics} 17:36, 8 September 2009 (UTC)
great input - if I was in a graduate class in topology this article works, but MOST are not — Preceding unsigned comment added by 75.163.175.56 ( talk) 18:16, 27 December 2015 (UTC)
See Stefan Bergman and Bergman kernel, not Valentine Bargmann and Segal–Bargmann space. Boris Tsirelson ( talk) 18:37, 10 May 2016 (UTC)
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 ? — Preceding unsigned comment added by 64.180.21.136 ( talk) 01:50, 29 October 2016 (UTC)
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.
It is not a exceptional case; much mathematics is developed with that other (hidden) convention.
It follows from properties 1 and 2 that a complex inner product is antilinear in its second argument, meaning that
Álvaro López de Quadros ( talk) 20:59, 1 March 2017 (UTC) Footnotes
The lack of precision in many mathematical texts of Wikipedia really is astonishing.
"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.
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 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. — Preceding unsigned comment added by 217.95.164.135 ( talk) 21:20, 29 September 2019 (UTC)
The Clarify-jargon template I added was summarily removed as a "drive-by tagging", whatever that is. So without re-adding it, let me explain why I put it there in the first place: I have no idea what Hilbert space is and I have tried to get through this article several times. Since I am a smart person who studied maths to A-level I conclude that my experience is not unrepresentative. To understand the first paragraph, you are asking readers to get a grip on: Euclidean space, vector space, complete metric space, mathematical series, absolute convergence, and norm (mathematics). It's just too much. My feeling is that you need something much more general:
...with further technical details coming in gradually lower down. The whole article has similar problems, however, which is why I tagged it. Widsith ( talk) 15:54, 8 September 2009 (UTC)
I have rewritten the first paragraph to address this concern. Sławomir Biały ( talk) 18:18, 8 September 2009 (UTC)
I've just discovered my book uses (f,g)=equation and <f,g>=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).
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. — Preceding unsigned comment added by 2601:143:480:a4c0:6dd5:ec40:6d14:9d00 ( talk) 10:04, 2 April 2020 (UTC)