Timeline for Barrelled, bornological, ultrabornological, semi-reflexive, ... how are these used?
Current License: CC BY-SA 2.5
32 events
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| Mar 10, 2017 at 9:42 | history | edited | CommunityBot |
replaced http://www.math.ucdavis.edu/ with https://www.math.ucdavis.edu/
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| Oct 19, 2012 at 12:13 | comment | added | Peter Michor | @Greg Kuperberg: In view of my answer, the place of convenient in this very nice diagram should be: Sequentially complete $\implies$ convenient. | |
| Oct 5, 2012 at 12:46 | comment | added | Abdelmalek Abdesselam | The nuclear property is fundamental for the construction of probability measures via the Bochner-Minlos Theorem. So it is important for probability and Euclidean quantum field theory. | |
| May 11, 2010 at 7:59 | comment | added | Andrew Stacey | I ended up leaving this open longer than I intended! Not that you need the reputation ... | |
| May 11, 2010 at 7:58 | vote | accept | Andrew Stacey | ||
| Dec 14, 2009 at 18:52 | comment | added | Andrew Stacey | (1) Point taken. I first met all this through "Convenient setting .." so my default is to accept whatever they said there. (2) I'm still hoping that more functional analysts will stop by and help divide these properties into "front line" and "supporting" roles - part of my purpose in asking this was to flush out more FAs to see if any have any ideas on my research questions (a scheme that has already born some fruit!). But I take your point there as well, and will only leave it open for a short time. | |
| Dec 14, 2009 at 15:00 | comment | added | Greg Kuperberg | (1) Convenient spaces are first defined in "Smooth structures", by Frolicher, where he clearly says bornological and locally complete. I think that Kriegl and Michor made a slight mess of things by both changing the definition and making a redundant term. (2) I know that you don't like to accept answers in general, but it is also possible to be at the bottom of a pile of unanswered questions. I think the answered pile is more useful to readers, and the software only refreshes questions with no upvoted answers. | |
| Dec 14, 2009 at 13:00 | comment | added | Andrew Stacey | With regard to accepting your answer (I know you were only joking, but nonetheless ...), I'm leaving it open in the hope that some (other) functional analysts stop by and also answer. I feel that people are more likely to look at an open question than an answered one and this is one of those topics where more points of view would be great. Given how many people have voted for the question as well as your answer, I think it would be a shame if it sinks to the bottom of the pile. | |
| Dec 14, 2009 at 12:57 | comment | added | Andrew Stacey | The subtleties of the relationships is one of the reasons why I want to transfer this to the n-lab: it'll be easier to keep track of it there and there's more scope to be inventive with the layout. Possibly a full database is the best, but I'm not sure I can be bothered with the labour. | |
| Dec 14, 2009 at 12:56 | comment | added | Andrew Stacey | In my default reference, after the definition of "convenient" the authors say "In [Frolicher, Kriegl, 1988] a convenient vector space is always considered with its bornological topology - an equivalent but not isomorphic category.". So I guess you're right, it is ambiguous. | |
| Dec 14, 2009 at 9:13 | comment | added | Greg Kuperberg | @Andrew: Also, the compound relations are more subtle than just equalities, for example nuclear Frechet implies Montel. This is one reason for a computer-assisted survey of these definitions. Another reason is to keep track of counterexamples, like Frechet AND Montel but NOT distinguished. That counterexample mindlessly implies Frechet AND reflexive but NOT distinguished, and that is where you want software to keep track. | |
| Dec 14, 2009 at 9:08 | comment | added | Greg Kuperberg | If you think it's fantastic, you should ACCEPT my answer, please. :-) Seriously, thank you for the praise, and for sure you can use these diagrams any time you want. Concerning your quibble, Google gives me inconsistent answers. One source says convenient = locally complete + bornological; another says it just means "bornologically" (or locally) complete. Who is more standard? m-hikari.com/ijma/ijma-password-2007/ijma-password13-16-2007/… | |
| Dec 14, 2009 at 8:51 | comment | added | Andrew Stacey | PS I was going to start adding some of this to the n-lab page(s) on LCTVS. May I use your diagram? Also, graphviz is great! Not only is it useful, it can lead to serious wastes of time: math.ntnu.no/~stacey/HowDidIDoThat/Random/sheffield.html | |
| Dec 14, 2009 at 8:48 | comment | added | Andrew Stacey | That is absolutely fantastic. Of course, it doesn't tell the whole story which is that often two lower properties are equivalent to a higher one (silly example being normed+complete=Banach, slightly more sophisticated being normed+nuclear=finite dim). Minor quibble: convenient is the same as locally complete. | |
| Dec 13, 2009 at 21:31 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
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| Dec 13, 2009 at 21:29 | comment | added | Greg Kuperberg | Yes, I used graphviz. | |
| Dec 13, 2009 at 21:28 | comment | added | Mariano Suárez-Álvarez | graphviz (graphviz.org) does that for you | |
| Dec 13, 2009 at 21:27 | comment | added | Qiaochu Yuan | What program did you use to write that diagram? I'd like to make a similar diagram and I don't want to have to draw it on paper since it would be difficult to change the structure of the diagram if there are too many inconvenient edges. | |
| Dec 13, 2009 at 21:23 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
Extended answer with Hasse diagram
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| Dec 11, 2009 at 17:17 | comment | added | Greg Kuperberg | I didn't mean that particular statement to be my personal opinion, so I changed it to something less categorical. | |
| Dec 11, 2009 at 15:28 | comment | added | Andrew Stacey | Oh, I didn't mean for you to do that! I think it was clear that your sentence was an opinion and not to be taken categorically (in the non-mathematical sense). I just wished to register the opposing opinion. (Since writing my thesis, I've come across a book on nuclear LCTVS by Pietsch which I think is a wonderful treatment of them.) | |
| Dec 11, 2009 at 8:36 | comment | added | Greg Kuperberg | Okay, I changed the wording some. | |
| Dec 11, 2009 at 8:36 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
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| Dec 11, 2009 at 8:31 | comment | added | Andrew Stacey | Very nice answer. I hadn't heard of the EDM, thanks for the reference. My only quibble is your tone when talking about nuclear spaces: I find their properties much more useful than those of Banach spaces - I wrote my thesis with a copy of Grothendieck's book on them next to my computer. | |
| Dec 11, 2009 at 7:54 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
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| Dec 11, 2009 at 7:45 | comment | added | Greg Kuperberg | Okay, I added that word. | |
| Dec 11, 2009 at 7:45 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
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| Dec 11, 2009 at 7:38 | comment | added | Mariano Suárez-Álvarez | The topology of a Fréchet space is defined by a countable family of seminorms (so that it is metrizable) | |
| Dec 11, 2009 at 7:27 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
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| Dec 11, 2009 at 6:49 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
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| Dec 11, 2009 at 6:32 | history | edited | Greg Kuperberg | CC BY-SA 2.5 |
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| Dec 11, 2009 at 6:25 | history | answered | Greg Kuperberg | CC BY-SA 2.5 |