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The first paragraph says: "The theorem is the justification for the bra-ket notation popular in the mathematical treatment of quantum mechanics."... What? I would hardly call it the "justification" for that notation (which can be easily explained in terms of inner products!); nor do I think that some
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This article seems well put together, but it would be nice if there was a "prerequisites" part that said what math courses would be appropriate for understanding the material. Even just an "Undergraduate" or "Graduate" designation would be helpful to minimize the amount of time needed by students to
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is a new editor for WP math articles: if you think you know this material well, and can improve it, then please do so. Just be careful not to wreck the article in the process: us "old timers" find many well-meaning edits to be of questionable value, often muddying or obscuring the subject matter.
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Kolmogorov-Fumin, Rudin), and the dual of L^p being L^q with 1/p+1/q=1 (e.g. Lieb-Loss, Royden). Although the opening text of this page is helpful, I think it would be useful to either have all three discussed on the same page or to have a disambiguation page where all three appear equally.
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There are three (or possibly four) results that are commonly called the "Riesz representation theorem": the dual of a
Hilbert space being itself (e.g. Lax, Reed-Simon), the dual of continuous functions being given by appropriate measures (with slightly different results for C_0 and C_c, e.g.
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By the way, I'm not really a new editor of math articles. I'm doing it more frequently now because I'm revising for exams, but also it hasn't been until recently that my browser would keep around the
Knowledge cookie for more than a few days, so most of my edits are anonymous.
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Hmm. This article, as currently written, is clear enough if one already knows the subject matter, and one's interest in reading it to "refresh one's memory". But if one does not know
Hilbert spaces well, then it appears confusing and poorly structured. I note that
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I think I understand now what the comment is attempting to say. I will move the comment lower down (because the theorem has applications outside of QM), and I will try to clarify it. If there is a problem, please fix rather than just reverting.
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It says, ISMW, that the distinction between real and complex forms is the appearance of the complex conjugation in the complex version. This doesn't fit the definition given for "anti-isomorphism" in the linked article.
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What is the source which says "Hilbert spaces are generally assumed to be complex"? This is not true for all fields of math. Often in, say, numerical analysis, one says
Hilbert space rather than real Hilbert space?
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Riesz representation theorem. Another textbook of mine, Funcional
Analysis by Michael Reed and Barry Simon, calls the first version the "Riesz lemma". (Disambiguation page is good, though.)
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two very different fonts are used for capital phi. one is hard to distinguish from lower case phi. This could be very confusing for people who don't already know what's going on.
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Those sound fine to me. And the page can always be moved later if you find a canonical name. (Lemma doesn't sound lame to me, but I didn't grow up speaking
Spanish.)
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imagined connection to physics is a helpful way to start the article. If anything, the sentence should be reworded and moved to the very bottom.
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Thanks. The edits looked good. I made a minor change. Also, my browser has a "remember me" button which keeps me logged in for weeks at a time.
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Everything, more or less, about
Hilbert space can be 'explained in terms of inner products'. I don't see that that invalidates what is said.
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Is the proof of the Riesz-Frechet representation theorem disregarding conjugate linearity? I suppose this works over
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Hmmm. "Riesz lemma" makes the result sound pretty lame (Riesz layma). We could give it a really long name.
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depend on the Riesz representation theorem for? Obviously more needs to be said by way of explanation.
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on
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This page should be split up into three articles. If there is no objection, I will do this soon.
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Er, not so happy about that. Rudin (Real and
Complex Analysis) presents the second version as
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I agree. What would you call the first two sections? The third one has a canonical name,
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is non-zero seems to rely on the Axiom of Choice. One fairly boring proof is via
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Should I have used the phrase "trivially explained"? I.e.
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650:{\displaystyle \mathbb {R} }
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