Talk:Hilbert space/GA1

GA Reassessment
This discussion is transcluded from Talk:Hilbert space/GA1. The edit link for this section can be used to add comments to the reassessment.

I intend to reassess this article in the next few days. I will add my comments to the table below. Geometry guy 20:41, 5 July 2008 (UTC)

Some examples from the text...
 * "the success of Hilbert space methods ushered in a very fruitful era for functional analysis." Did it? Or was the success of the methods part of the era?
 * "they are simply transformations that stretch the space by different factors in mutually perpendicular directions." In the lead, but not covered in the article.
 * "In a Hilbert space, the fundamental objects are abstractions of vectors, whose nature is unimportant" Misleading.
 * "While the definition of a Hilbert space given below may appear complicated, due to a large number of consistency axioms, the basic intuition behind Hilbert spaces is amazingly simple" The definition below isn't given by any axioms. Also the last statement is (standard) opinion and needs a cite.
 * "The Hilbert space theory of partial differential equations, in particular formulations of the Dirichlet problem." This can be read in two ways.
 * "Older books and papers sometimes call a Hilbert space a unitary space or a linear space with an inner product, but this terminology has fallen out of use." Are unitary spaces complete? Linear spaces with an inner product need not be, and neither the term "linear space" nor "inner product" have fallen out of use!
 * "Sobolev spaces, denoted by Hs or W s, 2, are another example of Hilbert spaces, and are used often in the field of partial differential equations." The grammar and the word "often" are awkward here, but the real question is why only one sentence on Sobolev spaces?
 * "Conversely, every Banach space in which the parallelogram identity holds is a Hilbert space, and the inner product is uniquely determined by the norm." How?
 * "Every Hilbert space is isomorphic to one of the form \ell^2(B) for a suitable set B." Said twice. Also, one should probably be clearer about the fact that this statement uses the axiom of choice.
 * "This allows to define its norm as... The sum and the composite of two continuous linear operators is again continuous and linear." Some copyediting required here.
 * "in fact, this is the motivating prototype and most important example of a C*-algebra." According to whom?

Please add comments below. I will update the above accordingly. Geometry guy 22:14, 6 July 2008 (UTC)
 * As improvement is taking place, I will wait at least a week (from the date of the review) before deciding whether to delist or not. I hope to have time myself to work on the article. Geometry guy 09:12, 7 July 2008 (UTC)
 * I'm away for a few days. I hope to revisit the article on Sunday. Geometry guy 22:20, 14 July 2008 (UTC)
 * There has been much improvement to the article. In particular, the combined introduction and history is a nice step forwards, and the applications section is much richer. There are still things to do. I will help if I have time, and revisit the article more carefully at the weekend, with a view to closing this reassessment. Geometry guy 22:46, 23 July 2008 (UTC)
 * Fantastic work, most notably by Silly Rabbit, has improved this article enormously, and it could easily be taken to FA status by dedicated editors. The lead is currently weak, but can easily be fixed once the article takes shape. However, GA issues still remain after many weeks (and I wish I had been able to help more): the weaknesses are obvious already on reaching the "Other" section (2.4) but there are further problems in later sections (e.g. "New Hilbert spaces from old"). Consequently I'm obliged to delist this article for now. Geometry guy 21:59, 29 July 2008 (UTC)


 * hello, Geometry guy. perhaps for the knowledgable folks who might contribute, exactly what are you suggesting be added regarding operators? (the article operator theory is nothing but a placeholder stub at the moment.) Mct mht (talk) 00:27, 30 July 2008 (UTC)
 * The material on operators has much improved since this reassessment began (it was non-existent at the time). However, there is still next to nothing on spectral theory, which is a major modus operandus in Hilbert space theory (e.g. what do physicist's do when they have an observable?). Instead we have a somewhat cryptic sentence "These operators share many features of the real numbers and are sometimes seen as generalizations of them." Unfortunately, Spectral theory is itself a rather weak article: spectral theorem may be a better place to look. Geometry guy 08:30, 30 July 2008 (UTC)


 * does one really wanna go there in this article? the spectral theorem is only a pre-departure point in contemporary spectral theory, for operator theorists. a proper overview would begin with, maybe, the commutant lifting theorem. and then there is analysis of the spectrum, "spectral theory" for non-OT functional analysts. in those contexts, the spectral theorem is elementary. including it doesn't necessarily convey the flavor of the subject to the reader of an article like this and putting too much emphasis on it might be somewhat misleading. also, analysis involving Hilbert spaces operators is obviously huge, and it is not clear what choices one should make in topics to allude to (why spectral theory and not other parts of operator theory? and operator algebras?), and in what depth for an article like this. Mct mht (talk) 20:52, 30 July 2008 (UTC)
 * The spectral theorem (in one form or another, typically for symmetric or hermitian operators) is surely taught in every major mathematics undergraduate program worldwide. As you say it is elementary and there is no need to emphasise it. However, mentioning it in a paragraph does not amount to emphasising it! Geometry guy 21:25, 30 July 2008 (UTC)
 * PS. Could you start a new thread on article talk if you want to take this further? Thanks.