Riesz Proves the Riesz Representation Theorem | Hacker News Reader

Riesz Proves the Riesz Representation Theorem

nonagon.org

76 pointstodsacerdoti14 comments

For folks wondering about applications of this theorem: it is a key building block in the theory of reproducing kernel Hilbert spaces (RKHS), which in turn are the building block of kernel support vector machines (kernel SVMs), which are widely used in machine learning applications.

The "kernel trick" from kernel SVMs only works because of the existence and uniqueness result from the RRT on the underlying Hilbert space.

There are several results called the Riesz representation theorem.

The article is about representing continuous linear functionals on a space of continuous functions as signed measures (or Riemann-Stieltjes integrals). This has lots of applications in ergodic theory or representation theory (e.g. disintegration of measures).

This result is essentially unrelated to the result characterizing continuous linear functionals on Hilbert spaces. It is also much more difficult to prove (the result on Hilbert spaces is rather simple).

If you can represent a continuous linear functional as the inner product with the Riesz representative then doesn't this also define a signed measure? It kind of seems like one of those theorems should imply the other to me, or is there some subtle aspect I'm missing?

It's the same thing really. It's just that Riesz first proved it for the special case and it was then generalized to Hilbert spaces. It's such a huge generalization that it causes a lot of confusion.

C([0,1]) is not a Hilbert space but a Banach space. Every Hilbert space is a Banach space, but not vice versa. The version of the theorem for Hilbert spaces is indeed a lot easier to prove than the one given in the article.

And that just proves my point about confusion :)

Thanks for the correction.

Integration is a linear functional on a vector space of functions. The RRT identifies the dual of continuous functions on a compact Hausdorff space with countably additive Borel measures. A very simple result is the dual of bounded functions on any set is the space of finitely additive measures. (E.g., https://keithalewis.github.io/tandon/vs.html) D. J. H. Garling has a very clever proof of the theorem based on this that is "short" if you know about the Stone-Cech compactification, the Hahn-Banach theorem, and the Carathéodory extension theorem. :-) https://www.cambridge.org/core/journals/mathematical-proceed...

Another source is W. Rudin, Real and Complex Analysis.

Well of course, and every other analysis text.

> Another source is W. Rudin

Isn't that the source with the proof described in the article as ungrokable?

I think the author takes issue with it not being intuitive, which is understandable. They do seem to acknowledge the generality is beneficial, however.

I went carefully, line by line, through the first, real, half of Rudin's Real and Complex Analysis.

All the writing I've read by Rudin is very precise. Sometimes a reader might want an intuitive understanding of what is going on, and for this after reading carefully take some time out, look back, and formulate some intuitive views. Right, in Rudin's books I've never seen a picture, but there is no law against drawing ones own pictures.

But for that subject, call it functional analysis, I also learned from Royden's Real Analysis, a little from each of several other books, and the lecture notes from the best course I ever had in school.

Overall, I liked learning from Rudin's books -- I'm glad to have such high quality math writing. But Halmos is my favorite author. And when they cover the same material, I like Royden better than Rudin. One of my main interests in that math is as background for probability, and for that my favorite author is Neveu.

Sorry about the OP: For me, the Riesz representation theorem is a very old topic; I covered it quite well in the past, don't want to go back, and am doing other things now.

For anyone who wants the Riesz theorem, in Rudin a nicely general version with a precise proof is on just a page or two with, say, a few more pages to get ready for the theorem itself.

Very lucid blogpost, but why does the author choose pseudo-anonymity?

What you mean? The post has the full author name at the bottom. And the front page of the site even has a eulogy for his wife that passed away, both people with full names and even what state he lives in. Pseudo-anonymity? What more do you want from the guy, his street address and phone number?

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