Timeline for Riesz representation for an infinite-dimensional space
Current License: CC BY-SA 3.0
13 events
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| Feb 4, 2013 at 18:28 | comment | added | Banach | @Yemon, which smaller space do you suggest to consider instead of $C(X)$? Do you know of a reference for Riesz representation for $C(X)$ where $X$ is a metrizable non LCH space? | |
| Feb 4, 2013 at 16:14 | comment | added | Yemon Choi | For instance, have you tried looking for versions of Riesz representation for $C(X)$ where $X$ is just a metrizable non LCH space? Or is the linear/affine structure of $X$ supposed to be important? | |
| Feb 4, 2013 at 16:12 | comment | added | Yemon Choi | @Banach: I am not misunderstanding the question. My point is that it is very odd to be purely considering the set of continuous complex-valued functions on $X$, it means that there is a lot of a structure that you are not using, and suggests that you should not really be looking at the dual of all of $C(X)$ but instead at the dual of something much smaller. Your initial emphasis on $X$ being a Banach algebra seems misguided and is most likely irrelevant to any characterization of the TVS dual of $C(X)$ that you seem to desire | |
| Feb 4, 2013 at 9:03 | comment | added | Banach | @Yemon, You are misunderstanding the question. I give you a Banach algebra $X$ that is infinite dimensional. I am asking for a topology on $C(X)$ and a characterization of its dual. I am asking if anyone has seen such sort of theorem. The focus is the infinite-dimensionality of $X$. Either you have seen a Riesz-type theorem for infinite-dimensional spaces or not. If you don't like $X$ being an algebra, assume it's just a Banach space, or assume it's just a topological vector space. | |
| Feb 4, 2013 at 2:24 | comment | added | Yemon Choi | Where are you using the Banach algebra structure on $X$? As it stands, I think this question is far too vague. What is your intended substitute for "Radon measure"? | |
| Feb 3, 2013 at 23:52 | answer | added | Alexander Shamov | timeline score: 3 | |
| Feb 3, 2013 at 22:50 | comment | added | Banach | $C(X)$ can be given a topology (most likely not a normable), but I don't want to specify one because that is part of the question. I am asking if there is some sort of Riesz representation theorem for an infinite-dimensional space $X$ where $C(X)$ has some nontrivial topology? | |
| Feb 3, 2013 at 22:30 | comment | added | András Bátkai | No, it does not have to be a Banach space. Which topology do you take then? | |
| Feb 3, 2013 at 22:29 | comment | added | András Bátkai | Which norm do you take to get a Banach space? Or what do you mean by $C(X)$? If it is the continuous functions on $X$, then without compactness they are not bounded. | |
| Feb 3, 2013 at 22:10 | comment | added | Banach | Is $C(X)$ suppose to be a Banach space? | |
| Feb 3, 2013 at 21:27 | comment | added | Banach | I didn't say $C(X)$ is a Banach space. | |
| Feb 3, 2013 at 21:24 | comment | added | András Bátkai | How do you make $C(X)$ a Banach space? | |
| Feb 3, 2013 at 21:10 | history | asked | Banach | CC BY-SA 3.0 |