Talk:Hilbert space

The section "bases" is reproduced verbatim in the article orthonormal basis. I assume this is done on purpose, but I don't see the reason for it. I do see a danger however: improvements in this article may not be incorperated in the original, and vice versa.

In the same section, I don't understand the sentence "only countably many terms in this sum [referring to the Fourier series] will be non-zero, and the expression is therefore well-defined". I suppose "well-defined" means that the sum is finite, but this does not follow immediately from the fact that the number of non-zero summands is countable. Am I misunderstanding the sentence? -- Jitse Niesen 15:17, 1 Mar 2004 (UTC)

for a generic vector space, only finite sums of vectors make sense. if the vector space has a norm, and is complete with respect to this norm, than you can take countably infinite sums of vectors, and use the norm to define a limit of this sum. if the partial sums form a Cauchy sequence, then this series is guaranteed to converge to a vector in the vector space (completeness). so countably infinite summations make sense, and might converge. none of this applies to uncountable summations.

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This article seems to assume a great deal more knowledge in the reader than can reasonably be expected. Obviously, when you're dealing with abstract mathematical concepts, it gets difficult to explain things in general terms, and without referencing other abstruse concepts and vocabulary. Still, I think Wikipedia can do a lot better than this (and it has-- see: Quantum Mechanics). After reading the article, I still had almost no idea of what a Hilbert space actually is. I can't imagine anyone who hasn't studied higher mathematics getting any use out of the article in its current form.

Thankfully, I managed to find a satisfactory explanation here. I think this too might be beyond what an average person can follow, but it's a good example of how to explain a mathematical concept in terms of physical phenomonon, and without reverting to mathematicianese. It also gives much more background to the Hilbert space than this article affords it, and which the subject deserves.

That page was written by Jack Sarfatti, a known crackpot. Many parts of his explanation are bad, in my opinion. Some parts are almost wrong, like this one, for example: "Hamilton formulated a new description of classical mechanics which was eventually housed in an infinite-dimensional phase space. In this space, a point represents the entire physical system." I would suggest not relying too heavily on Sarfatti's writings.

as far as this page goes, well, I suppose a certain amount of linear algebra and analysis are prerequisites for understanding Hilbert spaces. Perhaps it is possible to write to a less experienced audience, but I don't think that Sarfatti provides a good model for how to do so. Lethe

It's one thing to say my article is not "good", it's quite another to say I am a "crackpot". I challenge Lethe to give even ONE specific example in which I wrote something about physics that is "crackpot". Making errors as all do is not same as being "crackpot". I use only mainstream physics. I do not say that relativity is wrong. I use relativity. Ditto with quantum theory. So what is Lethe talking about? BTW:

Do you mean "Hamilton formulated a new description of classical mechanics which was eventually housed in an infinite-dimensional phase space. In this space, a point represents the entire physical system."

What's wrong with that? I am not saying phase space is Hilbert space. Is that what you assumed? In any case that statement hardly makes me a "crackpot". The phase space of classical fields is infinite. Jack Sarfatti JackSarfatti 00:12, 28 October 2005 (UTC)!

To understand Hilbert spaces, undoubtedly, but to explain what they are, what they are used for, and why they are significant, in a way that most people can follow? I don't doubt the mathematical explanations in this article are complete and accurate, and they should remain for those that can understand them. But encyclopedias need to be written primary for a lay-audience, not math majors. Where Sarfatti succeeds, regardless of the veracity of his article, is in framing the subject in a general context, and elucidating it in terms and analogies that are reasonably accessible. Wikipedia should strive for those qualities. I would love to be able to help improve the article myself, but obviously I don't understand the subject terribly well, so I must leave that job to those who do.

I'm not sure that I agree with you that all encyclopedia articles should be aimed at lay people. For example, the encyclopedia needs to have an article on the monster group, but I don't think there is any sensible way to explain what the monster group is to someone who doesn't know some group theory. Can you convince me that monster group should be aimed at the lay person? or how about Kähler manifold? that article is necessarily only readable by somewhat more mathematically experienced reader. In these cases, it is probably not even worth trying to aim the article to a layman, since the layman would most likely have no interest in these subjects.

The hilbert space article is somewhat similar, there are prerequisites to knowing a hilbert space that are simply unavoidable. Nevertheless, if you can suggest which parts are hard to follow, and why, I might try to help. Lethe | Talk

Hilbert spaces are different because they are linked from a lot of articles on quantum mechanics, making its relations to quantum mechanics more understandable to people who might have an interest in quantum mechanics but who perhaps only know basic calculus and linear algebra, like many chemists for example, would perhaps not be a bad idea. Passw0rd

It seems if we are going to talk about Hilbert spaces in relation to the QM, then we should, at some point, include a link to "Rigged Hilbert Space" since that is what is actually referred to in QM.

von Neumann

As I found it, the page on Hilbert spaces was incorrect to speculate that the abstract spaces were invented by Weyl in 1931. I have corrected the page to cite the 1929 paper in which von Neumann coined the term.