[HN Gopher] Riesz Proves the Riesz Representation Theorem
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Riesz Proves the Riesz Representation Theorem
Author : todsacerdoti
Score : 46 points
Date : 2021-10-02 11:27 UTC (1 days ago)
(HTM) web link (nonagon.org)
(TXT) w3m dump (nonagon.org)
| graycat wrote:
| Another source is W. Rudin, _Real and Complex Analysis_.
| ajkjk wrote:
| Well of course, and every other analysis text.
| [deleted]
| ur-whale wrote:
| > Another source is W. Rudin
|
| Isn't that the source with the proof described in the article
| as ungrokable?
| graycat wrote:
| 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.
| colossal wrote:
| I think the author takes issue with it not being intuitive,
| which is understandable. They do seem to acknowledge the
| generality is beneficial, however.
| alpineidyll3 wrote:
| Very lucid blogpost, but why does the author choose pseudo-
| anonymity?
| codetrotter wrote:
| 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?
| ajtulloch wrote:
| 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.
| hilber_traum wrote:
| 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).
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(page generated 2021-10-03 23:00 UTC)