Talk:Hilbert space/GA2

GA Review

Article (edit | visual edit | history) · Article talk (edit | history) · Watch

Review by Jakob.scholbach

I think by and large this is a well-done article which is close to GA status. Since I feel the article has potential, I'm giving a more thorough (and may be more demanding) review than usually the case. I'll try to relate my comments to the GA criteria more closely when I've read all the article.

• The lead section looks fine. It may be good to get a total outsider to see what they get from it. The lead does contain a lot of terms that are only understandable to a mathematically trained reader.
  • I'm not sure about the wording "number of dimensions". AFAIK the dimension can be a number (incl. infinite)? Is this standard English?
  • "In this connection" sounds odd
  • The sentence mentioning von Neumann reads a bit odd. Perhaps move J.v.N. to the beginning of the phrase?

I think I have copyedited to address these three points. Let me know if this resolves these issues. Sławomir Biały (talk) 01:31, 11 September 2009 (UTC)

• Intro and history
  • I would strongly suggest recalling the dot product on R³.
  • It is sometimes vague ("certain limits", "including series" ??, "can naturally be thought of"). Try to be as concise as possible.

The original goal of the introductory section was to be an accessible introduction, also combining the history. In this spirit, it would not do to say that a Hilbert space is a complete metric space. However, I am increasingly inclined to agree with you that what is needed here is for a brief definition, possibly supported by two or three simple examples, should precede the history section. Sławomir Biały (talk) 02:48, 11 September 2009 (UTC)

All right. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • I acknowledge that this is a general problem of history sections, but much of the material is hardly digestible at this point, e.g. "to prove an anlaog of the spectral decomposition".

This is probably less of an issue now with the new article structure. Sławomir Biały (talk) 13:35, 11 September 2009 (UTC)

All right. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • "just continuous functions" is misleading. Riemann integral can also integrate non-cont. fcts.

Fixed. Sławomir Biały (talk) 01:40, 11 September 2009 (UTC)

• • I don't have the time to check the references, but from a quick glance it seems that most of the references you give are original works. This is good, but claims like "Von Neumann was perhaps ..." have to be supported by some secondary material.

There is a general reference to Bourbaki, intended for most of the history section. But your point is taken that some specific statements are strongly-worded enough to require a copyedit and/or reference. Sławomir Biały (talk) 01:40, 11 September 2009 (UTC)

OK. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • Why does history end in 1930?

Funny. Actually, the Bourbaki reference, to which most of the history section is sourced I believe, goes only to the 1930s. Many of the histories in other books have this problem as well. A conceivable reason is that the history bifurcates rather dramatically at that time. I'm guessing there was an explosion of activity in many different areas when the notion caught on. I've now included a paragraph on Gelfand, Naimark, Segal, since that is arguably part of the "main line" on what should be included in a history of the subject, but that only gets us to the mid 1940s. There is, of course, the area of noncommutative geometry that could easily be mentioned in connection with von Neumann algebras, which is obviously of substantial contemporary importance. Sławomir Biały (talk) 05:48, 11 September 2009 (UTC)

OK. It is now extended a bit. I think for GA it's good enough, but that would be a spot to work on in the future. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• Applications:
  • I wonder why you put it this early. At this point, we don't even yet know what HS are. Consequently many terms are vague and/or difficult to understand, e.g. "candidate for a weak formulation".

I have moved it to later now. That may not totally settle the issue of terms being difficult to understand. Sławomir Biały (talk) 04:17, 11 September 2009 (UTC)

I think the later the better. But that's obviously not a GA criterion. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • Another general point: I feel the section is unbalanced in that it is fairly strongly centered on examples. You give one ex. for Sturm-Liouville theory and one for PDE's, but I suspect both are vast fields that need to be summarized somewhat more globally. I sense that devoting that much space to one example is too much, having in mind the total length of the article.

The example for PDE's is fairly characteristic, I think. The article shouldn't dwell on specifics of what a "weak formulation" is for an elliptic boundary value problem. It's easier to digest with a familiar example. The Lax-Milgram theorem needs to get mentioned, and the article gets there with maximum economy, although sacrificing a lot of generality. I've added a few sentences at the end of the section to indicate the generalizations. Sławomir Biały (talk) 05:05, 11 September 2009 (UTC)

Hm. I'm not totally convinced, but won't argue. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • Image captions tend to be (sometimes too) long.

I've shortened all of them. Maybe some of them are now too short? Anyway, my feeling is that almost everything important is already explained in the accompanying text. Sławomir Biały (talk) 02:33, 11 September 2009 (UTC)

Yes, it is the text that should (and now does better than before) explain, images "only" illustrate. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • "cannot ever hope" -- unencycl. language

Copyedited. Sławomir Biały (talk)

• • "energy surface" not defined nor explained

Hopefully addressed. Sławomir Biały (talk) 02:33, 11 September 2009 (UTC)

• • This may be just me, but the bullets (e.g. at "If Ut is a (strongly continuo...)" look ugly/superfluous.
  • What does the tilde in $f(\theta )\sim \sum _{n=-\infty }^{\infty }a_{n}e^{2\pi in\theta }$ mean?

Not worth explaining. Killed it. Sławomir Biały (talk) 02:33, 11 September 2009 (UTC)

• • "Whereas Fourier analysis decomposes a function defined on a compact set into discrete vibrations of a violin string" -- odd conjunction of mathematical abstraction + colorful imagination (IMO).

Copyedited. I'd like to keep the "vibrations" in there, because of the intuition for why the spectrum should be discrete, but I've moved it to a parenthetical remark. Sławomir Biały (talk)

Better now. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • Perhaps add a ref. for hearing the drum's shape?

Kac is now referenced. Sławomir Biały (talk) 02:33, 11 September 2009 (UTC)

• • Merge the "Other" section with introduction to the section (or expand to a proper section).

Killed it. Group representations are already mentioned in the introduction section. Something should be written about square-integrable stochastic processes, but good references seem to be lacking. Sławomir Biały (talk) 02:33, 11 September 2009 (UTC)

• • I think you should devote a section on quantum theory, given that you state "that it [the idea of HS] offers one of the best mathematical formulations of quantum mechanics."

This is worth doing. I will start one. Sławomir Biały (talk) 02:33, 11 September 2009 (UTC)

I have added a short section. I may add one or two more paragraphs. Sławomir Biały (talk) 12:15, 11 September 2009 (UTC)

Good. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• Definition
  • I would definitely put this as the first section. Accordingly, I would also suggest merging the introductory example into this section.

Done. Sławomir Biały (talk) 05:13, 11 September 2009 (UTC)

• • associating a complex scalar --> complex number

Fixed. Sławomir Biały (talk) 04:17, 11 September 2009 (UTC)

• • On my up-to-date Mozilla browser, I can't the brackets in text mode in this section.

Well... WP:ACCESS isn't a WP:GAC. It's too bad that my own preferred notation seems to cause such silly technical problems like this. I predict that some less suitable notation will need to be used, but this is not really a priority at the moment. Sławomir Biały (talk) 04:17, 11 September 2009 (UTC)

• • linearity misses "where a and b are in C"

Fixed. Sławomir Biały (talk) 04:17, 11 September 2009 (UTC)

• • Somehow I miss the mention that HS are topological vector spaces right up front. This may be trivial, but I feel it's important to evoke the notion of nearness etc. For example, later "it is a closed subspace" -- you link closed, which is good. However, I feel this is a central notion that might be worth explaining in this article.

It is central. I have followed the definition with this. I hope you don't find it out of place here. Sławomir Biały (talk) 04:17, 11 September 2009 (UTC)

Good. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • Perhaps add an image for the triangle inequality?

Done. Sławomir Biały (talk) 12:38, 11 September 2009 (UTC)

• • "For example, Cⁿ ..." misses "is a Hilbert space"

Section has been transmogrified, and the observation is now moot. Sławomir Biały (talk) 05:13, 11 September 2009 (UTC)

• • It may be worth saying a word about the sum over $b\in B$ if B is not N etc.

Done. Sławomir Biały (talk) 04:17, 11 September 2009 (UTC)

All right. I wonder, is there any subarticle one could link to? Perhaps it is worth creating Orthogonal direct sum?

• • "For example, Let ..." --> "Let ..."

Done. Sławomir Biały (talk) 05:15, 11 September 2009 (UTC)

• • what is the meaning of "however" in "The full Lebesgue..."

Copyedited. Sławomir Biały (talk) 04:58, 11 September 2009 (UTC)

• • "Sobolev spaces [...] are Hilbert spaces [...] retain all the useful geometric properties of Hilbert spaces" is redundant
  • "which relying more ..." wording

Copyedited stray word. Sławomir Biały (talk) 04:58, 11 September 2009 (UTC)

• • "consisting entirely" -- I don't understand the meaning of "entirely"

Copyedited. Sławomir Biały (talk) 04:58, 11 September 2009 (UTC)

• • In direct sum: σ, λ etc. are not defined

The section should just summarize the material from the spectral theorem anyway. I've gotten rid of the explicit formulas, so there is now no need to define these. Sławomir Biały (talk) 13:24, 11 September 2009 (UTC)

• • do not embolden anything except the absolute most important terms. e.g. not internal direct sum

I think I have removed all errant emboldened terms. Sławomir Biały (talk) 04:58, 11 September 2009 (UTC)

• Properties
  • the permutation invariance is somehow oddly worded. Perhaps "... is independent of the order of the vectors"?

Fixed. Sławomir Biały (talk) 14:18, 11 September 2009 (UTC)

• • What is the importance / role of Bessel's inequality?

I have attempted to address this by moving Bessel's inequality to the section on orthonormal bases, where it can be given better context alongside Parseval's formula. I am a little worried that I might now be disrupting the flow of that section, and it would be helpful if someone else could take a look at it and comment on it. Sławomir Biały (talk) 13:05, 13 September 2009 (UTC)

• • "One has a complete" -- unencycl. wording. Also, that sentence should come after the definition, I guess.

Fixed. Sławomir Biały (talk) 04:58, 11 September 2009 (UTC)

• • Is H* a Hilbert space again?

Yes. Now the article mentions this explicitly. Sławomir Biały (talk) 04:58, 11 September 2009 (UTC)

• • I wondered why "Duality" section is not a subsection of "Operators"

That's so the article can discuss Weakly convergent sequences. Also, the Riesz representation theorem is often treated alongside the other fundamental properties of a Hilbert space. Certainly, the article could be restructured in another way, but I think it is good where it is now. Sławomir Biały (talk)

I don't insist, but the first few lines of the two sections being basically identical strikes me as odd. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • What does "restoring linearity" mean?

I've attempted to clarify this. u_φ is antilinear in φ, so the reversal of order "restores linearity". Sławomir Biały (talk) 04:58, 11 September 2009 (UTC)

• • "A Hilbert space is, first and foremost, a Banach space" -- well, then, why is it so late that we learn about that?

I've fixed this issue by mentioning that they are Banach spaces just after the definition. I have also removed this sentence. Indeed, the Banach space properties are probably some of the least frequently used properties of Hilbert spaces, simply because the techniques that hold in Hilbert spaces are so much more powerful. Sławomir Biały (talk)

All right. Perhaps the point that you just made here might appear somewhere in the article. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • In the same section, it is odd that you formulate results that hold for Banach spaces only for Hilbert spaces. Somehow consider rewording (such as in the remark about the different quality of proofs).

Fixed, except for the geometrical Hahn-Banach which I think is just best left in a Hilbert setting. Sławomir Biały (talk) 04:58, 11 September 2009 (UTC)

OK. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

I'll read the rest tomorrow or Saturday. Jakob.scholbach (talk) 22:37, 10 September 2009 (UTC)

• I like it better with the definition up front. However, the definition of dot product now appears twice. Except for this it is a real improvement!

Does it appear twice? I may have lost track of things, but I did get rid of the one definition I was aware of in preparing this section. Sławomir Biały (talk) 11:34, 12 September 2009 (UTC)

Well, the R^3 subsection mentions the three properties of inner products and the definition repeats it. Perhaps skillfully trim that down somewhat? Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• Orthonormal bases
  • Use bold folt sparingly.

Done. Sławomir Biały (talk) 11:34, 12 September 2009 (UTC)

• • The point "if ⟨v, ek⟩ = 0 for all k ∈ B and some v ∈ H then v = 0." somehow should be connected to "A subset S of H spans a dense vector subspace if (and only if) the vector 0 is the sole vector v ∈ H orthogonal to S.", I guess.

Nice observation. I will do that. Sławomir Biały (talk) 11:34, 12 September 2009 (UTC)

• • "Note that" is unencycl. (in general, any addressing the reader should be avoided)

Done. Sławomir Biały (talk) 11:34, 12 September 2009 (UTC)

• • The "Hilbert dimension" section redundantly repeats material from before. Merge that.

I have moved the discussion of l^2(B) for B a set into the orthonormal basis section, since this is the only place in the article where the general definition is used. It also gives a ready example of spaces with different Hilbert dimensions. Sławomir Biały (talk) 12:20, 12 September 2009 (UTC)

• • Maybe state more expressis verbis that Φ is an isomorphism of Hilbert spaces?

I'm not sure I understand you. Surely that is what the article already does? Sławomir Biały (talk) 11:43, 12 September 2009 (UTC)

Sorry about that. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • "the natural state space of boson seems to be" -- whose opinion is that. Provide a reference.

This entire example is a paraphrase of the Streater and Wightman reference. I'll work on the referencing so that this is clearer. Sławomir Biały (talk) 11:43, 12 September 2009 (UTC)

Better. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • "The orthogonal projection PV is a self-adjoint linear operator ..." -- self-adjoint appears only later (and is not explained at this point). This suggests reordering the material. Likewise for the notation P^∗ .
  • The equation with the operator norm of projectors: it might be good to delineate more clearly that/why this is a special feature of projections (as opposed to general cont. maps). Also, the operator norm is not defined at this point.
  • I would somehow trim the paragraph "The orthogonal complement satisfies ..."

It isn't my favorite paragraph either, even though I wrote it. It was my attempt to give some flesh to some isolated results that had been scattered throughout the article without much attempt to organize them. I'm not sure what to do with it. Perhaps the whole paragraph should simply be removed, and the remark about Hahn-Banach moved (somehow) into the Banach space properties section. Sławomir Biały (talk) 12:26, 12 September 2009 (UTC)

This would perhaps fit nicely in the article Orthogonal direct sum (to be created) or something like that? Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• Operators
  • "Conversely, if an operator is bounded, then it is continuous." -- this should appear right after the other implication (before the definition of the norm)

Fixed. Sławomir Biały (talk) 13:05, 13 September 2009 (UTC)

• • In the discussion of self-adjoints, the spectrum is not defined yet. Since you talk about spectrum later, this should be moved down.

It should be mentioned, but maybe just imprecisely, in the section on self-adjoint operators. I've put something in about self-adjoint operators being "real", without really saying what that means. Sławomir Biały (talk) 14:10, 12 September 2009 (UTC)

Aha. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)

• • " If T is self-adjoint, then the spectrum is real. " -- redundant (appears above)

I think this is ok now. Sławomir Biały (talk) 13:05, 13 September 2009 (UTC)

• • "spectral value" not defined

Fixed. Sławomir Biały (talk) 13:05, 13 September 2009 (UTC)

• • nor is "compact operator" (OK, the link is there, but a brief info would be nice)

I defined compact operators, Fredholm operators, and outlined a few of the reasons they are important. Sławomir Biały (talk) 14:10, 12 September 2009 (UTC)

• • The "spectral theory" section has not a single reference.

Added references. Sławomir Biały (talk) 13:05, 13 September 2009 (UTC)

• • "partial order defined on self-adjoint operators" not defined/explained

Clarified. Sławomir Biały (talk) 14:10, 12 September 2009 (UTC)

• • "An unbounded operator ... whose domain D(T) is a subspace of H" -- are there any specifications? I mean, it must always be a (linear) subspace, if the operator is linear, right?

The word "linear" was missing. Sławomir Biały (talk) 14:10, 12 September 2009 (UTC)

• • If "the spectrum of an unbounded is defined in precisely the same way as for bounded operators" I suggest that you make an extra section on spectrum etc. and branch into bounded/unbounded later.

Let's now check the criteria

1. Well-written (grammar etc): OK
2. Factually accurate and verifiable: accurate OK, verifiable almost OK (see above queries for references)
3. Broad in its coverage. (a) it addresses the main aspects: without being a connoisseur of Hilbert spaces, I think it does address most "classical"/"main stream" aspects of the topic. So this should be OK for a good article. (b) it stays focused on the topic without going into unnecessary detail: looks reasonably well-done. (I do think, though, when aiming for a more comprehensive article (i.e., FA), you won't have the space for lots of technical and/or more basic stuff. Accordingly, this should be deferred to subpages. For example, spectral theory could accommodate quite a bit of content. It might also be worth thinking about creating Glossary of Hilbert spaces, akin to Glossary of topology, Glossary of scheme theory etc.)
4. Neutral: OK
5. Stable: OK
6. Images : OK.

So, in conclusion, I think I'm going to promote the article to Good Article status once the issues above have been addressed in the same way as the first half already has been. I'll put the article on hold for a few days (or more if you need) and check back then. Jakob.scholbach (talk) 10:52, 12 September 2009 (UTC)

Some other general things (that are not part of the GAC and thus are just suggestions at this level):

• I have to say that the article is at times close to a textbook, which it should not. I get the impression that parts are essentially what would be written in any standard book (without having ever looked into any, admittedly).
• I feel that most of the material is what guys into Hilbert spaces and functional analysis may call elementary/easy.
• The global article organization is often not good. It is not easy to read the article from beginning through end without jumping. I guess it is not completely avoidable, but I'd like to urge you to rethink the general structure -- what are the interdependencies, what are the central organizing principles etc. Currently it is a bit a list of facets that are not (optimally) intertwined and therefore the reading is at times a bit ragged and also quite technical and so momentwise a bit boring (to me). For example, the section on Hardy spaces is well-done in itself, but contains almost only the definition. Instead, it should, IMO, be a summary of Hardy space (or an ideal version thereof). Also, except for the one mention in Bergman spaces not at all linked to the rest of the article. Jakob.scholbach (talk) 10:52, 12 September 2009 (UTC)

---

Thanks, Sławomir Biały, for so swiftly adressing the comments above. The more important issues above are covered, so I think I'm promoting the article to GA class. Jakob.scholbach (talk) 15:11, 14 September 2009 (UTC)