Talk:Riesz representation theorem

Split up article

This page should be split up into three articles. If there is no objection, I will do this soon.CSTAR 02:23, 24 Dec 2004 (UTC)

I agree. What would you call the first two sections? The third one has a canonical name, Riesz-Markov theorem. Dbenbenn 03:42, 24 Dec 2004 (UTC)

How about Riesz representation theorem and Extension of Radon measures? There could also be a disambiguation page called Riesz theorems or something.CSTAR 03:53, 24 Dec 2004 (UTC)

Er, not so happy about that. Rudin (Real and Complex Analysis) presents the second version as the 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.) Dbenbenn 04:27, 24 Dec 2004 (UTC)

Hmmm. "Riesz lemma" makes the result sound pretty lame (Riesz layma). We could give it a really long name. Riesz representation theorem for linear functionals on Hilbert spaces or Riesz representation theorem on Hilbert space. CSTAR 04:35, 24 Dec 2004 (UTC)

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.) Dbenbenn 05:50, 24 Dec 2004 (UTC)

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 find books to explain the prerequisite knowledge. —The preceding unsigned comment was added by 151.203.160.33 (talk) 06:13, 8 January 2007 (UTC).

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. 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. — Preceding unsigned comment added by 129.215.104.198 (talk) 17:28, 7 August 2014 (UTC)

Introduction

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 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. A5 16:34, 17 February 2006 (UTC)

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. Charles Matthews 16:49, 17 February 2006 (UTC)

Should I have used the phrase "trivially explained"? I.e. $\langle a|b\rangle =\langle a,b\rangle$. What do the operator $\langle a|\equiv \langle a,\cdot \rangle$ and the vector $|b\rangle \equiv b$ depend on the Riesz representation theorem for? Obviously more needs to be said by way of explanation. A5 18:36, 25 February 2006 (UTC)

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 User:A5 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. linas 16:15, 26 February 2006 (UTC)

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. A5 20:13, 27 February 2006 (UTC)

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 Wikipedia cookie for more than a few days, so most of my edits are anonymous. A5 20:44, 27 February 2006 (UTC)

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. linas 01:18, 28 February 2006 (UTC)

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? --LavaCircus (talk) 02:20, 28 March 2023 (UTC)

minor font issue

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. Looksurly (talk) 19:59, 20 September 2013 (UTC)

Misplaced "anti"?

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.

Perhaps it should say "anti-isometric" instead?

198.228.228.161 (talk) 02:44, 6 February 2014 (UTC) Collin237

conjugate linearity in the proof of Riesz-Frechet

Is the proof of the Riesz-Frechet representation theorem disregarding conjugate linearity? I suppose this works over $\mathbb {R}$, but shouldn't $f={\bar {\phi (g)}}g$ if we want to generalize to the complex case?

Proof is incomplete

The proof is incomplete because it doesn't show that $M^{\perp }\neq \{0\}$ when $\varphi \neq 0$. I detected there was something wrong when I couldn't find any usage of the completeness of $H$ or the continuity of $\varphi$ in the argument. --Svennik (talk) 09:09, 28 January 2021 (UTC)

The statement that $M^{\perp }$ is non-zero seems to rely on the Axiom of Choice. One fairly boring proof is via Zorn's lemma. --Svennik (talk) 13:43, 28 January 2021 (UTC)

I've added a few sentences to the proof explaining that showing that $M^{\perp }$ is non-empty relies on Zorn's Lemma, the Cauchy completeness of $H$, and the continuity of $\phi$. Nowhere else in the proof are these assumptions used. --Svennik (talk) 14:14, 28 January 2021 (UTC)