Timeline for Hahn-Banach theorem with convex majorant
Current License: CC BY-SA 3.0
29 events
| when toggle format | what | by | license | comment | |
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| Apr 8, 2020 at 8:58 | comment | added | Martin Sleziak | The formulation with convex $p$ is also given in Aliprantis-Border (page 195, Theorem 5.53), as mentioned in my question on Mathematics: Reference for the range of possible values in Hahn-Banach Theorem. | |
| Feb 13, 2020 at 23:43 | answer | added | Piotr Hajlasz | timeline score: 2 | |
| Jul 17, 2018 at 23:31 | comment | added | Pietro Majer | Now the geometrical form, in the particular case of subgraphs of convex/concave functionals and graphs of affine functionals produces the extension you mention. So from this point of view, the functional form with "convex functional" instead of "sublinear" should appear as a half-way generality. | |
| Jul 17, 2018 at 23:31 | comment | added | Pietro Majer | I'm not sure, but I suspect, that the "geometrical form" (separation of convex sets by means of hyperplanes) of HB is older than the "functional form", or at least, it has been considered prior with respect to to the latter. I mean, I suspect the functional form was initially intended as a lemma or method to prove the general statement about the geometrical form. | |
| Jul 17, 2018 at 23:06 | answer | added | MichaelGaudreau | timeline score: 10 | |
| May 24, 2018 at 17:14 | comment | added | Jochen Wengenroth | @Evan Aad No, you need a real valued majorant. Otherwise the result is wrong. | |
| May 24, 2018 at 16:52 | comment | added | Evan Aad | When you wrote: 'However the theorem is true if the majorant $p$ is merely convex', did you also have in mind convex functions that take $\infty$, or only real-valued convex functions? | |
| Nov 18, 2015 at 9:11 | comment | added | Jochen Wengenroth | @DelioMugnolo That every convex function $p:\mathbb R\to \mathbb R$ can be represented as $p(x)=\max \lbrace f(x): f$ affine-linear and $f \le p\rbrace$ is not completely trivial but follows from Hahn-Banach with convex majorant after reducing to $x=0$ and $p(x)=0$. | |
| Nov 18, 2015 at 0:31 | comment | added | Delio Mugnolo | > "the result is even interesting for X=R" I cannot understand this remark. If $L=\{0\}$, then the assertion (that there exists a linear function below the convex function) is trivial, and even more so if $L=\mathbb R$. Or am I overlooking something? | |
| Nov 2, 2015 at 19:24 | answer | added | dalry | timeline score: 5 | |
| Oct 28, 2015 at 14:25 | vote | accept | Jochen Wengenroth | ||
| Oct 28, 2015 at 11:00 | comment | added | Christian Clason | I think some extensions in this direction are given in Stephen Simon's From Hahn-Banach to Monotonicity. | |
| Oct 28, 2015 at 1:43 | answer | added | fedja | timeline score: 34 | |
| Oct 27, 2015 at 14:30 | comment | added | Mateusz Wasilewski | The only reference I have for this are lecture notes from the functional analysis course I took 5 years ago (in Polish, it is "Twierdzenie 3" on page 3): mimuw.edu.pl/~torunczy/AF1/Wyk+cw10-11/AF10.pdf I can write an e-mail to professor Toruńczyk to find out, if he has a more precise reference. | |
| Oct 27, 2015 at 11:17 | comment | added | Jochen Wengenroth | @MateuszWasilewski The separation theorem (which is, in some sense, a version of Hahn-Banach) is due to Mazur. But I did not find the extension version with convex majorant attributed to him. | |
| Oct 27, 2015 at 9:39 | comment | added | Jochen Wengenroth | @DuchampGérardH.E. Of course. I have edited the question. | |
| Oct 27, 2015 at 9:38 | history | edited | Jochen Wengenroth | CC BY-SA 3.0 |
Replaced $P|_X$ by $p|_L$.
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| Oct 27, 2015 at 0:05 | comment | added | Danu | This question seems like it would (also) be a good fit for History of Science and Mathematics. @Jochen, what would you think of a migration? | |
| Oct 26, 2015 at 19:51 | comment | added | Jochen Wengenroth | @MateuszWasilewski This is a good hint. I will try to check it. | |
| Oct 26, 2015 at 19:11 | comment | added | Christian Remling | @BillJohnson: OK, I have to confess: I'm really interested mainly in (spectral theory on) Hilbert spaces. The incriminating evidence is here: www2.math.ou.edu/~cremling/teaching/ln.html | |
| Oct 26, 2015 at 18:59 | comment | added | Bill Johnson | I don't understand, Christian. If you are dealing with normed spaces you need the separation theorem, which more or less forces you to prove the subllnear version. | |
| Oct 26, 2015 at 18:43 | comment | added | Christian Remling | It is not mandatory or advisable to always extract the most abstract and general version of the statement from a given proof. Maybe people didn't bother with convex because sublinear is completely adequate for what they want to do with HB (when I teach a FA class, I prove HB for normed spaces only because this is all I need later). Just my 5c though. | |
| S Oct 26, 2015 at 16:51 | history | suggested | Ali Taghavi |
I add two tags
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| Oct 26, 2015 at 16:21 | comment | added | Ali Taghavi | @Jochen What are some applications which can not be proved with the "sublinear" version? | |
| Oct 26, 2015 at 16:18 | review | Suggested edits | |||
| S Oct 26, 2015 at 16:51 | |||||
| Oct 26, 2015 at 15:49 | comment | added | Mateusz Wasilewski | I heard that this version is due to Mazur but I don't have any reference at hand. | |
| Oct 26, 2015 at 15:10 | comment | added | Dirk | Wow, you know more than 100 books on functional analysis! | |
| Oct 26, 2015 at 14:15 | history | edited | Willie Wong |
edited tags
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| Oct 26, 2015 at 13:07 | history | asked | Jochen Wengenroth | CC BY-SA 3.0 |