Talk:Hilbert space/Archive 1

message: It's one thing to say my article is not "good", it's quite another to say I am a "crackpot". I challenge Lethe to give even ONE specific example in which I wrote something about physics that is "crackpot". Making errors as all do is not same as being "crackpot". I use only mainstream physics. I do not say that relativity is wrong. I use relativity. Ditto with quantum theory. So what is Lethe talking about? BTW:
:I didn't claim that you were a crackpot because the article was wrong.  I claimed that you were a crackpot and the article was wrong.  The implication was that if you are a crackpot, of course your articles about Hilbert spaces will be wrong. -Lethe | Talk 02:20, 28 October 2005 (UTC)
replies: text: These are the mistakes in that sentence, as I see them.  First, Hamilton never formulated classical mechanics in infinite dimensional phase space, nor did his description eventually become housed in infinite dimensions.  Hamiltonian mechanics is formulated in phase space, which is 6n dimensional for n particles in 3 dimensional space.  This is finite.  Second, in both quantum mechanics and classical mechanics, finite dimensional or not, a point in the phase space represents only one possible state of a physical system, not the entire system.  The physical system is capable of other configurations, which are not represented by a point in phase space.  -Lethe | Talk 02:20, 28 October 2005 (UTC)

text: I was going to complain about ad hominem attacks, but I suppose I started this whole debate with an ad hominem against you in the first place, by calling you a crackpot.  So go ahead, attack my credentials and my motives.  -Lethe | Talk 02:20, 28 October 2005 (UTC)
replies: text:  Whoa! OK, I understand  you don't like being called a crackpot. But Lethe made that comment over a year ago, and I think some sort of statute of limitations would apply to off-the-cuff remarks. And I'm one of those arrogant mathematicians, so please I feel offended. Give a link to your side of the story about Baez.  That would help.--CSTAR 13:02, 28 October 2005 (UTC)
replies: text: A little googling turned this up, though I was in grammar school when that was written, so I'm not sure what it has to do with me (other than that I'm a fan of Baez' This Week's Finds). -Lethe | Talk 00:55, 5 November 2005 (UTC)
thread_title:  Sarfatti's defense of his Hilbert space article 

message: I've noticed that nearly all articles of a scientific or mathematical nature on wikipedia are nearly indecipherable to the kind of person who would most likely be referencing the subject on wikipedia. While I'm sure the article is well-researched, I doubt that the kind of person who could decipher most of what's in this page would be looking it up on Wikipedia. 
Now, that being said, it is true that part of the wikipedia concept is that it should be a storehouse of knowledge. I'm therefore not suggesting that anything in this or other articles of a similar nature should be deleted; as a "record" of knowledge it seems to stand pat (though I'll let the experts dispute that if they want, I have no idea myself, I'm just speaking in terms of content style); but there should be an effort to include a seperate "layperson's" section in these types of articles that explains the basic concepts to the person most likely looking up this type of article. In this case it is most likely somebody who is trying to learn about qunatum physics and has stumbled across the term "hilbert space", has no idea what it means, and goes to wikipedia to unsuccesfully find clarity. 
And, no, I'm not volunteering, because I'm a layperson on this subject myself. I've certainly endeavoured to do this in my area of speciality. If somebody wants to test their teaching skills, I'd love for somebody to decrypt these types of things. (Unsigned comment from User:Sayfadeen 17 January 2006).
:Yes, this article starts out a bit too abstract, and should provide a simple example, probably before the formal definition. In the meantime, the best two simple examples are the discrete Fourier transform, and the orthogonal polynomials. Once you understand those, perhaps this article will become accessible. linas 06:06, 18 January 2006 (UTC)
replies: text: It's pretty hard to draw pictures of things that are 4-dimensional.  It's inconceivable to draw pictures of things that are infinite dimensional. -lethe talk [ +] 21:15, 2 February 2006 (UTC)

text: Go lethe!  linas 00:59, 3 February 2006 (UTC)
replies: text: 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 - 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. Sojourner001 23:05, 16 January 2007 (UTC)
thread_title:  The Layperson 

message: "If a linear operator is defined on all of a Hilbert space then it is necessarily bounded. However, if we allow ourselves to define a linear map that is defined on a proper subspace of the Hilbert space, then we can obtain unbounded operators."
I've deleted this, http://en.wikipedia.org/wiki/Discontinuous_linear_map gives a general example of unbounded operator on any infinite dimensional normed space into any non-zero normed space, so this is not true. —The preceding unsigned comment was added by Scineram (talk • contribs) .
: That should have read "if a closed linear operator is defined" etc.  I'll fix it.--CSTAR 14:57, 27 February 2006 (UTC)
thread_title:  this is wrong 

message: "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)
replies: text: You have not guessed right.  The criterion for Cauchy convergence of a sequence {xi}i is that ||xi–xj|| get small.  This is a norm of differences.  If instead we asked that the difference of norms got very small, what would happen with an infinite sequence of independent unit vectors?  All their norms are 1, so all differences of norms are 0, but this sequence does not converge (and so should not be Cauchy). -lethe talk [ +] 01:24, 7 March 2006 (UTC)
replies: text: Very true, but how is the difference of two Hilbert space vectors defined?  Loodog 03:48, 7 March 2006 (UTC)
replies: text:   x-y = x + (-1) \cdot y, \,  --CSTAR 03:52, 7 March 2006 (UTC)
replies: text: I guess the problem is I'm trying to contexualize this in L2.  What would the difference of f(x)=x^(3/2) and g(x)=exp(-x^2) be?Loodog 04:07, 7 March 2006 (UTC)
replies: text: Well, I'm not sure what your domain is, if it's L2(R), then the first function is no good: it's not square integrable.  If your domain is something smaller, [0,1] for example, then you should be OK.  Anyway, in any function space, sums, differences and scalar multiples are defined pointwise, so the difference is just x^(3/2) – exp(–x^2). -lethe talk [ +] 04:14, 7 March 2006 (UTC)
thread_title:  Rewording "norm of differences" 

message: 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)
replies: text: I guess one of the problems with bra-ket notation is that you often write a ket with an eigenvalue as its label, and for some operators, there are points in its spectrum for which there are no vectors in Hilbert space.  So over-reliance on eigenvector-eigenvalue correspondence is one problem.  I guess the other is that it obscures the distinction between a Hilbert space and its dual.  Even though it's OK to do so because of the Riesz lemma, I think mathematicians prefer to maintain the distinction. -lethe talk [ +] 17:21, 9 March 2006 (UTC)
replies: text: Off-topic, maybe, but .. coming from a physics background, I eventually learned that not all operators are trace class, and this was a shock and a big conceptual problem I'm still wrestling with.  Worse, one actually does encounter non-trace-class operators in physics. There's a certain naivete in physics that can get you into trouble when one starts getting into harder material. linas 01:54, 10 March 2006 (UTC)
thread_title:  frowning on bra-kets 

message: "Since all infinite-dimensional separable Hilbert spaces are isomorphic"
Isomorphic as what?  And are separable and inseparable spaces isomorphic? —The preceding unsigned comment was added by 59.144.16.174 (talk • contribs) .
:Isomorphic as Hilbert spaces.  An isomorphism of Hilbert spaces is an invertible linear isometry.  A separable Hilbert space is not isomorphic to a nonseparable Hilbert space.  This is the Hilbert space analogue of the statement from linear algebra that two vector spaces are isomorphic if and only if they have the same dimension.  -lethe talk [ +] 08:07, 24 April 2006 (UTC)
thread_title:  What does this mean? 

message: 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)
replies: text: I have heard that statement often.  It's true, but pretty stupid.  Anyway, I'll try to add something about the topology to the article. -lethe talk [ +] 23:24, 30 June 2006 (UTC)
replies: text: Well I added some stuff to the article which might help.  Let me know if it does.  -lethe talk [ +] 23:53, 30 June 2006 (UTC)
thread_title:  Hilbert Spaces and Topology 

message: Could someone please revise the Definition section of this article in the following ways:
:Set the actual definition off from the introductory material (probably a separate paragraph; perhaps without a pronoun reference to 'this norm').
:Clarify the distinction between a Hilbert Space and a Banach Space.  Both articles mention a (cauchy convergence) complete normed space.  The Hilbert Space Definition mentions that inner products give rise to a norm but doesn't make it clear and explicit that the norm must be the inner product norm that is mentioned at the top.
:Move the examples of the space to the examples section.
--DrEricH 23:09, 21 August 2006 (UTC)
thread_title:  Definition section revision request 

message: Firstly, I would like to mention that I am not a mathematician. The last time I covered Hilbert spaces was in an undergraduate Linear Algebra course over 20 years ago and I only understood the concept in the context of quantum mechanics. Therefore I will not comment on content, but on editorial standards and style. In my opinion, this is a Good Article, however, I would like to make a few suggestions for improvement:
*In the mathematics Manual of Style, it is suggested to include “an informal introduction to the topic, without rigor, suitable for a high school student or a first-year undergraduate”. Perhaps just one or two sentences is required in the introduction.
*The applications of Hilbert spaces (quantum mechanics, PDEs, stochastic processes) should be in the “Introduction” section rather than the “Definition” section.
*It is mentioned that Dirac notation is frowned upon by mathematicians. Having “grown up” on the notation (in the field of physics), I was surprised to read that. If it is stated in the article here, for completeness, it may be wise to state why that is the case. The article on Dirac notation unfortunately doesn’t mention why (probably because it was written by physicists).
*The section “Operations on Hilbert spaces” has only one sentence. Can this be put into another section or expanded?
*As noted in the mathematics Manual of Style, the use of the first person (“we”) should be avoided.
*I would suggest giving a list of basic references, under a “Further reading” appendix section.
I have placed the article “On hold” for the changes to be implemented. RelHistBuff 10:03, 10 October 2006 (UTC)
replies: text: I tried to implement most of these suggestions and make some other basic changes. I think the page could still be improved quite easily. I didn't make too much progress on the informal introduction and didn't add any basic references. I did try to clean up the language a little bit and remove some unnecessary chaff. –Joke 15:01, 10 October 2006 (UTC)

text: The article looks better now. For a basic reference suggestion, I thought about the textbook I used, but I can't recall the title and in any case it's probably out-of-print. A short search turned up this possibility: Theory of Linear Operators in Hilbert Space by Akhiezer and Glazman. The reason I thought this one might be good is that it is a Dover publication, one of my favourite publishers for good, basic, low-cost, and classic math and physics textbooks. Probably the experts here know of better or more examples. RelHistBuff 09:40, 11 October 2006 (UTC)
replies: text: I came by again and although it is still a day early from the hold limit, I promoted the article to GA. I would still recommend adding a "Further reading" appendix with a list of basic references. However that does not stand in the way that this is a good article. RelHistBuff 07:42, 16 October 2006 (UTC)
thread_title: Good Article nomination

message: 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)
thread_title:  Definition 

message: 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)
thread_title:  Image problem 

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

message: 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)
thread_title:  Inclusion of Folland reference 

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

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

message: 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)
thread_title: Good work here

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

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

message: 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)
replies: text: Thanks, I had forgotten this earlier discussion.  It seems I was wrong in assuming that ⟨ and ⟩ would resolve to the correct character just because they are named html entities.  I've gone back to Ybunalobill's version before the introduction of LaTeX.  Sławomir Biały (talk) 11:16, 30 August 2011 (UTC)

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

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

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

message: 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)
replies: text: This all seems very off topic in a general article on Hilbert spaces. Sławomir Biały (talk) 19:55, 27 November 2013 (UTC)
replies: text: As far as I know, the very first (now rather naive, probably) form of the no-cloning theorem was (mathematically) just the fact that the map  H \ni x \mapsto x \otimes x \in H \otimes H  is not unitary. This is about Hilbert spaces, but mathematically too trivial for being mentioned here. And the dagger compact category (and all that) is too far from the general theory of Hilbert spaces. Boris Tsirelson (talk) 21:08, 27 November 2013 (UTC)
replies: text: 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.  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 Talk:Quantum teleportation (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 POVM could be cut ... if for no other reason than the POVM article is horrid. Happy Thanksgiving! And again, excellent job on this article! User:Linas (talk) 16:10, 28 November 2013 (UTC)
thread_title:  no cloning theorem 

message: "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)
thread_title:  Harmonic rather than overtone 

message: In the article, the general expression is given as
:\bigoplus_{i\in I}H_i
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)
replies: text: 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 sum \sum_{i\in I}\|x_i\|^2 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 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).  Sławomir Biały (talk) 14:26, 12 April 2014 (UTC)
replies: text: Yes, you are, of course, correct. I read the section (and also Conway) too quickly and missed the point. YohanN7 (talk) 15:07, 12 April 2014 (UTC)
thread_title:  Direct sum 

message: How do complex affine spaces relate to complex Hilbert spaces? — Cheers, Steelpillow (Talk) 12:32, 30 September 2015 (UTC)
thread_title:  complex affine spaces 

message: I don't have time for a full blown GA review, but I do have some comments that may need to be addressed before GA is passed.
*My first comment is about the accessibility of the lead. I fear that you will have lost most of your potential readership by the end of the first paragraph. Most engineers and physicists do not know what a Cauchy sequence is, and will only have a very vague notion of the concept complete. To make the article accessible without a background in mathematics, the first paragraph needs to explain complete in more every day terms. One option (that makes me and probably every mathematician here cringe): "... is complete, meaning that if every converging sequence of vectors has a limit within the space." Of course, using the term converging in this way is not really mathematically correct, but it does a much better job at transferring the idea at an intuitive level. But I will be the first to concede that this option also isn't optimal. There must be a better solution.
*The article claims that almost all Hilbert spaces in physics are separable. This is not true pretty much all Hilbert spaces occuring in quantum field theories (= most of modern theoretical physics) are not separable. (They are generated by an orthonormal basis |p1,p2,...>, where the pi's are the momenta and thus continuous parameters.)
*The article makes frequent use of ⟨ and, which do not render on my work PC, Which runs a quite standard XP install. Is it possible to find an option that works on more PCs?
Maybe I'll comeback for a more detailed pass later. (TimothyRias (talk) 08:26, 8 September 2009 (UTC)
replies: text: Another option: "... is complete, meaning that it is not a dense subspace of a larger space." Boris Tsirelson (talk) 10:48, 8 September 2009 (UTC)

text: About nonseparable spaces: are you sure? The usual representation of a free field (be it bosonic or fermionic) is in the separable Fock space. Or do you mean something essentially non-free? As far as I understand, a physicist can write |p1,p2,...> not because he really needs nonseparability, but rather because on his, somewhat heuristic level, it is usual to treat the continuum as just a very fine lattice. Boris Tsirelson (talk) 10:53, 8 September 2009 (UTC)
replies: text: Well, supposedly the |p1,p2,...> are supposed to be orthogonal to each other of the pi's don't match. So, unless you know a way to make an uncountably infinite number of mutually orthogonal vectors in a separable Hilbert space this means that the Hilbert space must be non-separable. But there might be way to formulate it more carefully in which the Hilbert spaces are separable. (If space is assumed to be compact for example.) My understand of this has always been that physicist simply don't care. (TimothyRias (talk) 11:50, 8 September 2009 (UTC)

text: The old version of the lead had "complete, meaning that if a sequence of vectors approaches a limit, then the limit is guaranteed to be in the space as well".  I will try this: "complete, meaning that if infinite summation of vectors converges absolutely, then it converges to some limit within the space."  This has the advantage of being technically accurate, and probably communicates more to engineers than the current version. Sławomir Biały (talk) 11:04, 8 September 2009 (UTC)
replies: text: I think that could work. To make it even more accessible you could also spell out converges absolutely to something like "... if the norm of an infinite summation of vectors converges than it must have a limit within the space." Or something along those lines. There may be a significant subset of potential readers that wouldn't know what absolute convergence is. (But does know what a series, limit and converging mean.) Anyway, your proposal already reaches many more readers than just "Cauchy". (TimothyRias (talk) 11:50, 8 September 2009 (UTC)
replies: text: We usually teach absolute convergence in freshman calculus, so I think it is reasonable to expect that most potential readers will have at least some familiarity with the concept, although probably not for vectors, and certainly not in infinite dimensions.  I would also resist trying to spell things out too much.  Too much detail is just as likely to make the lead inaccessible as too little.  The lead could say "converges absolutely (in norm (mathematics))" instead of just "converges absolutely", at least to give the term "absolute convergence" some more meaningful context.  However, the term "norm" is more likely to be unfamiliar than "absolute convergence", I reckon, and taking the time to spell it out is likely to make the description of the space too long.  Less is more.  Sławomir Biały (talk) 12:17, 8 September 2009 (UTC)
replies: text:  Although absolute convergence is usually taught in most physics bachelors it is one of those subject which most (physics) students ended up forgetting again. (simply because it is a concept they rarely use.) Vectors on the other hand are the bread and butter of many applications and are thus likely to remain fresh in mind. I agree however that the lead should also not expand too much in fear of becoming unreadable. Your suggestion of the (in norm) addition will probably do the trick for most people. (TimothyRias (talk) 12:32, 8 September 2009 (UTC)

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

text: Actually, Fock spaces have little to do with it. The Hilbert space of single particle state in QFT is already nonseparable. Considering multiple paticle states actually makes this even worse. This simply comes down to the fact that [a(p),a†(p')]=(2π)3δ3(p-p') => |p> and |p'> are orthogonal if p!=p'.(TimothyRias (talk) 12:03, 9 September 2009 (UTC)
replies: text: No. In the same way you could say that the Hilbert space of the quantum mechanics is nonseparable. Indeed, it is rather usual (among physicists, not mathematicians) to use spaces |x> concentrated at points x. Of course, their scalar product is also a delta-function, and so, they are orthogonal. However, all that is not about separability, it is about a physicist's way to use delta-function as if it was a function. In fact, the state |x> is non-physical in many aspects: it is of infinite moment, of infinite energy etc. (Not that the mean energy is infinite, but much worse: energy is infinite with probability one.) Physical states are square integrable functions. Square root of delta-function is not. And the same holds in the quantum field theory. At least if you do not leave the Fock space. Physicists often say they do leave, but when they are serious about it, they mean another separable space. Boris Tsirelson (talk) 15:26, 8 September 2009 (UTC)

text: Also, Timothy, do not forget to sign... Boris Tsirelson (talk) 15:30, 8 September 2009 (UTC)

text: Let me add: it is common knowledge that (at least in the framework of a quantum theory of free bosonic massive field) that the one-particle sector of QFT is the same as the Hilbert space of quantum mechanics (of a single particle). (On the level of kinematics I mean, not dynamics.) If in doubt, ask Physics project. Boris Tsirelson (talk) 15:34, 8 September 2009 (UTC)
replies: text: Yes your right. I got confused by the bad habits of physicists. (TimothyRias (talk) 12:03, 9 September 2009 (UTC)
replies: text: Nice. Happy editing. Boris Tsirelson (talk) 13:48, 9 September 2009 (UTC)

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

message: See Stefan Bergman and Bergman kernel, not Valentine Bargmann and Segal–Bargmann space. Boris Tsirelson (talk) 18:37, 10 May 2016 (UTC)
replies: text: 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.  Sławomir Biały (talk) 19:13, 10 May 2016 (UTC)
thread_title:  Bergman, not Bargmann 

message: 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)
replies: text: Well spotted. I have corrected it. I hope the original artist doesn't mind. — Cheers, Steelpillow (Talk) 09:02, 29 October 2016 (UTC)
thread_title:  Superposition Image Problem 

message: 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)
thread_title:  Lack of precision 

message: The Clarify-jargon template I added was summarily removed as a "drive-by tagging", whatever that is.  So without re-adding it, let me explain why I put it there in the first place: I have no idea what Hilbert space is and I have tried to get through this article several times.  Since I am a smart person who studied maths to A-level I conclude that my experience is not unrepresentative.  To understand the first paragraph, you are asking readers to get a grip on: Euclidean space, vector space, complete metric space, mathematical series, absolute convergence, and norm (mathematics).  It's just too much.  My feeling is that you need something much more general:
:A Hilbert space is an abstract spatial concept used by mathematicians to consider the relations between certain types of vectors (quantities which have both magnitude and direction).  As such, it is one of a number of different kinds of vector spaces, and is named after David Hilbert, who helped develop the idea.  While many other vector spaces are concerned with a two-dimensional plane, or three-dimensional space, a Hilbert space extends such ideas into an infinite number of dimensions.  They are an indispensible tool in many areas of mathematics, physics, and engineering.
...with further technical details coming in gradually lower down.  The whole article has similar problems, however, which is why I tagged it. Widsith (talk) 15:54, 8 September 2009 (UTC)
replies: text: Nowhere in WP:JARGON, WP:MTAA, or WP:LEAD does it say that the lead should be so watered down as to say virtually nothing about the subject, as the above paragraph does.  On the contrary, the lead is supposed to define and give context the subject, and to summarize the contents of the article.  The current lead does this fairly thoroughly, I think.  While it is perhaps not accessible at a level where someone without any mathematical background (indeed, would such a person even have a use for an article about Hilbert spaces?), it truly does not overindulge in technical details.  The present first paragraph attempts to introduce Hilbert spaces in the simplest possible way, and offers up an additional sentence to say why both the inner product and completeness are important.  Sławomir Biały (talk) 16:49, 8 September 2009 (UTC)
replies: text: You ask what need someone without a mathematical background would have of this article.  The need I have, being such a person I suppose, is that I want to know what a Hilbert space is and I expect an encyclopaedia to tell me.  If you are writing this for an audience already versed in the subject, then it seems to me you've missed the point of the project.  If you think this is the "simplest possible way" to introduce the subject, I can only say I don't believe you.  Widsith (talk) 17:06, 8 September 2009 (UTC)
replies: text: For instance, you have objected to the use of the word "completeness" in the lead.  This is actually a fundamental aspect of the Hilbert space, just as important as the "linear space" aspect of the theory.  So, if you want to understand Hilbert spaces, you also need to understand the implications of completeness.  Rewriting the lead so as not to mention this fundamental property is a major omission, and will certainly not help anybody to understand the concept.  From WP:MTAA: Do not "dumb-down" the article in order to make it more accessible. Accessibility is intended to be an improvement to the article for the benefit of the less-knowledgeable readers (who may be the largest audience), without reducing the value to more technical readers.  Sławomir Biały (talk) 17:11, 8 September 2009 (UTC)
thread_title:  Jargon 

message: 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)
thread_title:  Unrelated contrivances in article