Timeline for bornological vector spaces over a non-archimedean field
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 16, 2019 at 19:17 | comment | added | arsmath | Just out of curiosity: in "also over any field with more exotic structures", what are the more exotic structures? | |
| Mar 4, 2013 at 13:10 | comment | added | Oren Ben-Bassat | Thanks very much, that was very confusing since there are other notions of discrete valuation (e.g. for the field $\mathbb{Q}_{p}$) that seem to completely conflict with their terminology. After looking more at the literature I have concluded that the notions of bornological spaces, or even quasi-complete bornological spaces make sense for trivially valued fields but the notions of complete bornological spaces do not. I think there should be a nice way to fix this though and give a better definition that works in all cases. | |
| Mar 3, 2013 at 13:46 | comment | added | Federico Bambozzi | Everything works over non-trivially valued fields of any kind. When Houzel says "non-discritely valued field" he really mean "non-trivially valued field". He is using an old-fashioned way of call a trivially valued field a "discrete valued field" since the trivial valuation equip the field with the discrete topology. I think that functional analysts uses this terminology sometimes, for example you can find it also in the GTM number 3 on topological vector spaces. | |
| Mar 1, 2013 at 18:02 | comment | added | Oren Ben-Bassat | I can see now that the assumption of being non-trivially valued is used. However, I really wonder how much of it works if we assume the field is non-trivially valued but do not assume it is not discrete. | |
| Feb 27, 2013 at 9:50 | vote | accept | Oren Ben-Bassat | ||
| Feb 27, 2013 at 8:46 | comment | added | Federico Bambozzi | Yes, because if you try to develop the theory of Mackey convergence in this settings you find that a sequence is convergent if and only if become stationary. The word bornivorous is stolen from "Bornologies and functional analysis" by Henri Hogbe Nlend, North-Holland Mathematics Studies 26; which is another good reference for the theory of bornological vector spaces and the duality with topological vector spaces. | |
| Feb 27, 2013 at 5:19 | comment | added | Oren Ben-Bassat | Dear Federico, Thanks for the great answer. I will probably accept it unless someone comes up with something in the next few days which could improve on that. Also, thanks for using the word bornivorous, what a great word! It seems that with the standard definition of completeness that every bornological vector space over a trivially valued field is complete. This seems a little strong but maybe it makes sense. | |
| Feb 26, 2013 at 20:11 | comment | added | Federico Bambozzi | My name is not displayed in the answer... so I am Federico Bambozzi :) | |
| Feb 26, 2013 at 20:08 | history | answered | Federico Bambozzi | CC BY-SA 3.0 |