Timeline for What is the name for a Banach space property closed under ultraproducts?
Current License: CC BY-SA 3.0
9 events
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| Sep 28, 2015 at 13:47 | comment | added | Jason Rute | Ok, @BillJohnson, thanks again! I see now that my initial characterization of super-property was missing the isometric embedding requirement. I've fixed that and also updated my question to include this requirement. (Notice, my two examples are closed under both ultraproducts and isometric embeddings.) | |
| Sep 28, 2015 at 13:39 | history | edited | Jason Rute | CC BY-SA 3.0 |
added additional assumption about being closed under isometric embeddings
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| Sep 27, 2015 at 21:43 | comment | added | Bill Johnson | Because any hereditary property possessed by all finite dimensional spaces and ultraproducts is possessed by all Banach spaces. Incidentally, James' original definition of super-Q was that every space that is finitely representable in such a space must have Q. In particular, the property must be hereditary. A space $Y$ is finitely representable in a space $X$ iff $Y$ embeds isometrically into some ultrapower of $X$. | |
| Sep 27, 2015 at 20:21 | history | edited | Jason Rute | CC BY-SA 3.0 |
separated constant C into two constants
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| Sep 27, 2015 at 20:20 | comment | added | Jason Rute | @BillJohnson, thanks for the correction! I spoke too soon. There is a constant $C$ that I forgot to account for. When that is accounted for, this property is preserved by ultraproducts. I fixed my question accordingly. However, I don't see why your example "explains why there is no name for properties that are preserved under arbitrary ultraproducts." Could you please elaborate? | |
| Sep 27, 2015 at 20:12 | history | edited | Jason Rute | CC BY-SA 3.0 |
added 1780 characters in body
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| Sep 27, 2015 at 15:27 | comment | added | Bill Johnson | Your last sentence is not correct. Finite dimensional spaces have cotype $q$ for all $q$, and every Banach space embeds isometrically into an ultraproduct of finite dimensional spaces. That explains why there is no name for properties that are preserved under arbitrary ultraproducts. Finite inequalities are approximately preserved under ultraproducts; see papers by Ward Henson to see how this elementary observation leads to a model theoretic characterization. | |
| Sep 27, 2015 at 3:42 | history | edited | Eric Wofsey |
edited tags
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| Sep 27, 2015 at 1:35 | history | asked | Jason Rute | CC BY-SA 3.0 |