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algebra and calculus from the two-dimensional
Euclidean plane and three-dimensional space to vector spaces of any finite or infinite number of dimensions. is simply false. A Hilbert space is a well-defined mathematical structure. It does not extend the methods of linear algebra and calculus rather these methods can also be applied to it. is correct. Given the structure of Hilbert space, one can apply/use/extend many methods from 2,3D linear algebra and calculus to obtain interesting results such as results about the distance of a point to a convex set. This, essentially my wording using <allow: -->
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function which, thus, can be
Lebesque-integrated over IR which yields 0. One has that chi_S is a so-called Null-function, i.e., it's a function that has support on a Lebesque-measure 0 set. In order to make the space L2 of sq-integrable functions a Hilbert space, one has to consider L2/N where one factors out the space N of Null-functions. This annoying, minor inconvenience is often "suppressed", i.e., everything is considered mod N really. ---- This should be mentioned, since without this modification (or philosophy of ignoring N), L2 is not a Hilbert space since it has no definite norm.
1333:, where this article was quite literally top of the list of potential GAR reassessments; I can see how some paragraphs are split by mathematical symbols, but at the same time you have entirely unsourced sections of pure prose (Probability theory, Color perception) and sections with mathematical lines which don't have any citations anyway (Pythagorean identity, Bounded operators, etc.) I appreciate that this is all "standard stuff" in textbooks, but if you could cite the standard textbooks that would be very helpful in not having to bring this to GAR.
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associated therewith. But to answer your concrete question: to verify the material here a reader would look at the linked sources, and turn to the relevant chapter or section of the source, and skim down to where the topic is addressed. This is not made significantly easier by a footnote for every line (though including one (1) hyperlink in each section of the article to the relevant chapter of a well written online textbook or scan of a paper textbook
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is that you avoid getting Good
Article credit getting taken away from some other long-past contributor. It doesn't help most readers much because the prose is generally in good shape, just badly sourced. A lot of the most problematic articles are in mathematics and there are very few active Good Article contributors in mathematics (mostly me although there are some others who have the expertise but usually contribute elsewhere). —
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1395:"sub-quality" seems like a largely arbitrary summary. Text that is easily verifiable and even has relevant sources linked at the bottom but doesn't have a footnote littered after every sentence is not really notably worse in quality or less useful to readers than text which does have such footnotes. Maybe "sub quality" could be defined as "spent insufficient effort checking bureaucratic tick boxes". –
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dispute, he first reverted my edit, almost without acknowledging its merits. While avoiding edit warring by himself, he actually incorporated in his 2nd revert-edit almost all of the concerns which I tried to fix with my initial edit of the first paragraph of the present article. Now, he takes credit for fixing these things.
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A footnote doesn't itself make text verifiable. A reader still has to understand the given source well enough to see that it does, in fact, verify the text it's supposed to. For an article like this, verifying any of the cited claims will require at least a year or two of university-level mathematics
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I agree that the sections you list are unsourced and need sources. But my past contributions to this article have been quite minor; it hasn't even been on my watchlist. I could probably handle some of these, and I'll add it to my (long) list of parts of
Knowledge to work on, but it would take quite a
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The lead says: "The mathematical concept of a
Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of
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has edited the first paragraph of the lead with the edit summary "Improved some
English (which was BAD). If one talks about "complete", then one needs to name the DISTANCE d, in this case, one needs to name that d comes from the inner product. This was incoherent before. Bad Math." In fact, the edit
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I agree that the sourcing could be better in places. I've tried to add sources any obvious place I could find. As others have observed, most of this content can be easily sourced to general textbooks. In fact, the biggest problem I have had is choosing from a number of textbooks which one is the
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be helpful, by saving the reader the trouble of checking a book out from the library). In the case of this article in particular (and many other technical articles covering basic parts of technical curricula) there are hundreds of sources containing more or less the same material; if you don't like
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is very keen to apply these standards retroactively and immediately to all past GAs as well, setting up a big cleanup effort to turn content formerly evaluated as good into sourced content or (possibly more likely) to decimate our listing of old Good
Articles. The only reward for doing this cleanup
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A Hilbert space is no methods (or meta-methods) which can be extended. . It extends the methods of linear algebra and calculus from the two-dimensional
Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. . It allows to extend the methods of linear
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Thanks for the summary, David; I'd like to add that the reason I'm not hanging about waiting for sections to be cited is that, as noted above, there are very few active contributors, and so any content which currently doesn't meet GA standards is likely to remain sub-quality for the next decade or
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A Hilbert space H has a norm coming from the inner product by definition. The norm must be definite, i.e., ||x||=0 iff x=0 also by definition. The one-point set S={0} c IR is closed in IR, hence
Lebesque measurable, and the characteristic function of S (say chi_S) is a positive Lebesque-measurable
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has acknowledged with his first revert-edit that a previous serious FLAW in the first paragraph of this article was that there was no connection made between completeness and the metric that comes with the inner product. I pointed that out in the header/summary of my first edit as he quotes above.
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from the inner product. So, I have reverted the edit. I was immediately reverted with an edit summary consisting essentially in irrelevant personal attacks. So, I have reverted again, with an edit summary containing links the the guidelines that apply here. By the way, I have fixed several other
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More careful reading shows that the sourcing is not as bad as it looks. Some sections have catch-all footnotes near the front like "See this text for this entire section, but really you can find all this in any good textbook on the subject", and these footnotes are correct. This is basic textbook
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I'm an Econ PhD. I too think the article needs a simpler section. For example, I cannot tell if
Euclidean space is a Hilbert space from the article as it currently stands. How about using Euclidean space as the first example? Show that it is a vector space, using the 8 criteria for that (inverse,
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From the article: "John von Neumann coined the term abstract Hilbert space in his work on unbounded Hermitian operators. Although other mathematicians such as Hermann Weyl and Norbert Wiener had already studied particular Hilbert spaces in great detail, often from a physically motivated point of
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removed my contribution in a haste without taking its merits into account as is obvious from the many elements of concern that I have put forward and that he incorporated in his 2 revert-edits. He takes credit for fixing points that I have raised and had already corrected as documented above.
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has previously removed contributions of other editors which were language-wise in order and also content-wise in order without being able to give a solid reason why he did so. This can be documented within WP and was verified with native speakers and mathematical experts. Also in the present
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think this perhaps it would be better to find something more useful to do with your time. I would say that the vast majority of effort spent on Knowledge is research and writing with the goal of conveying meaningful explanations to readers, and discussions (and sometimes conflict resolution)
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One difference might be that a Hilbert space can be over the complex numbers. But does that really do anything other than double the number of dimensions? And in any case, evidently there's already an extension of Euclidean spaces that includes complex dimensions: affine spaces.
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consequence of the definition of an inner product, and I feel that it's more accurate to say therefore that the latter "induces" the former. It's a small thing, I know, but I wanted to set out my reasons properly, and the edit comment seemed like a bad place for all this detail.
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addition, etc.)--- that is start from the idea of a set, not of a vector space. Then show that dot product has the needed properties of multiplication. Then show completeness (which is where I get lost for Euclidean space-- but others will have gotten lost earlier).
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For those contributors: The context is that Knowledge's Good Article process recently made its sourcing rules more strict, requiring all text that is not a summary of later sourced content to have an inline footnote, no later than the end of the same paragraph.
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in a sentence. . In mathematics, a Hilbert space (named for David Hilbert) generalizes the notion of Euclidean space. . In mathematics, the term Hilbert space (named for David Hilbert) generalizes the notion of Euclidean space. Speaking of <a Hilbert space:
1178:, as many important properties of Euclidean spaces cannot be extended to Hilbert spaces. Now fixed with the formulation "Hilbert spaces allow generalizing the methods of linear algebra and calculus from ... to ...", which fixes also the second concern of
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Probably—after all, the entirety of Knowledge is a bureaucratic exercise in ticking boxes. Then again, I don't see how such text is verifiable if it doesn't have a footnote after it, and just gestures vaguely at a mass of sources. Many thanks to you and
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My recent edit was quite minor. I agree, though, this looks under-referenced. Other than that it appears from a superficial look to be in pretty good shape, though. It needs effort to bring it back into GA shape but I think it should be possible.
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Consult the book W.Rudin "Functional Analysis". The so-called dual Banach space H* of a Hilbert space H (space of all cont. lin. functionals on H) can be identified with H in that any F in H* can be written as F(x)=<y|x:
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So an important question for the current article to answer is what makes a Hilbert space (a term unfamiliar to many readers of this page) anything other than a Euclidean space (a topic familiar to a broader audience)?
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used after it was mentioned that the inner product allows for a distance to be defined, the connection between inner product (which defines a canonical distance as in 3D space) and completeness was definitely made.
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1121:, then this means in English while talking math (Oxford dic) that one speaks of a distance which arises naturally in a given context (according to a known formula). With that wording <canonical distance: -->
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A Hilbert space is a Banach space and that important fact should be mentioned at the top of any article dealing with it, as it allows Banach space techniques to be applied such as duality and weak compactness.
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solved this by complicating the wording, introducing a symbol for naming the Hilbert space, and calling "vectors" its elements. Using vector here is not convenient, as, in many Hilbert spaces, typically in
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Greetings! I've noticed you've made an edit on this article. Does it still deserve its GA status? I thought it lacked inline citations, and was about to affix the template when I noticed the rating...
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Please, do not forget to sign your posts in talk page with four tildes (~~~~). This would avoid to other editors the boring task of doing it for you. I have also indented your post as recommended in
1098:) Hilbert space generalizes the notion of Euclidean space. There are earlier versions of this article which seem to concur with this. I found the old version bad English using the article <a: -->
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is blatantly false for the purpose of being right after the fact. Had he understood my first edit, he wouldn't have simply deleted my version, but built on my editing in a constructive way.
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in this context)". "Perpendicularity" is here because I believe that, for many readers, it is less technical than "orthogonality". If I am wrong, "Perpendicularity" could be removed.
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the concept of a Hilbert space — and in particular the infinite-dimensional version — was first published. That way, people searching for that information will be able to find it.
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dimensions. A Hilbert space is a vector space equipped with an inner product, an operation that allows defining lengths and angles. Furthermore, Hilbert spaces are complete, "
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and technical terms (canonical, complete metric space, Banach space). The edit did not fix what it was supposed to fix, namely, that it must be clear that the distance
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are an important generalization of Hilbert spaces, but mentioning them in the first paragraph is unhelpful for readers that know them and confusing for others.
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A mathematical structure (old version) or a term (LMSchmitt's version) cannot generalize a notion. Now formulated as "Hilbert spaces allow generalizing ..."
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background. Consequently, a reader who can use the citations at all won't need them after every sentence, and probably not for every paragraph. What's the
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870:-- " define a Euclidean space as a set of points on which acts a real vector space, the space of translations which is equipped with an inner product"
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of footnoting each paragraph in a section with "Chapter 12 of Smith (1980)", "Chapter 12 of Smith (1980)", "Still in chapter 12 of Smith (1980)"...?
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Introducing a letter H did simplify a sentence significantly. In my opinion, it was useful and the simplification it caused made it necessary.
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is speaking of a mathematical object (Ex.: Incidentally, a Hilbert space is complete.). Not a notion. So, the notion of (or <the term: -->
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I'm not sure if you were intending to support or contradict the user's point, but in that passage you quoted, no date is mentioned.
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AFIK, "angles" are generally not considered in Hilbert spaces, except for right angles. Therefore, I have replaced "angles" by "
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best to cite. If there are any other places that anyone thinks are insufficiently sourced, let me know.
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The old article itself pointed to "complete metric space" as the current version does. So why bashing me.?
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bit of time, maybe a month, if I'm doing it all myself. Perhaps more active recent contributors such as
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for all x in H where y=y(F) is also in H. In that sense with conjugate linear identification H*-: -->
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There's only one paragraph in the Sturm–Liouville theory section (and it already has a footnote).
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The edit did not fix what it was supposed to fix, namely, that it must be clear that the distance
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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more. Also, there's no prejudice against you taking GA credit for the article—no one's checking!
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That's a very nice problem to have. As for other places where an inline citation would be nice:
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that when using "distance", it must be clear between what the distance must be considered. But
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It is wrong, or at least confusing, to say (both versions) that Hilbert spaces generalize
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Similarly for the Direct sums and the Bessel's inequality and Parseval's formula sections
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a distance function". "Define" is technically correct, but the connotations are off. A
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the sources specifically linked, just pick up any textbook about the topic. –
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view, von Neumann gave the first complete and axiomatic treatment of them."
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Can someone provide information about the self-duality of Hilbert spaces?
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Space of square integrable functions is not a Hilbert space as claimed.
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is something that a human imposes, whereas the relevant metric is an
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The Hilbert space has (or potentially has, depending on definition)
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So, the current version fixes some issues that were pointed out by
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Here are some flaws of the old version that are badly corrected in
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entirety of Knowledge is a bureaucratic exercise in ticking boxes
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impropernesses of the formulation that I did not remark before.
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See section "Duality", where the matter is discussed at length.
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The first and third paragraphs of the Probability theory section
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y(F), H becomes its own Banach space dual, i.e., self-dual.
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651:This was a
606:Mathematics
597:mathematics
546:Mathematics
148:free images
31:not a forum
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1589:definition
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1433:XOR'easter
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