Timeline for Relative compactness... but what is the toplogy?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Apr 24, 2021 at 10:29 | comment | added | edamondo | Thank you for the references. I think it is clear now | |
| Apr 23, 2021 at 19:15 | comment | added | Dirk Werner | @edamondo: In addition to Jochen's hint, the classic "Vector Measures" by Diestel and Uhl is a source to look at. | |
| Apr 23, 2021 at 0:55 | comment | added | Jochen Glueck | @edamondo: A reference for the duality in case that $X$ is reflexive is, for instance, Corollary 1.3.22 in "Hytönen, van Neerven, Veraar, Weis: Analysis in Banach Spaces, Volume I (2016)". In fact, what one needs is not necessarily reflexivity of $X$, but the weaker property that the dual space $X'$ have the Radon-Nikodým property; see [op. cit., Def 1.3.9, Thms 1.3.10 and 1.3.26, and Def 1.3.27]. | |
| Apr 22, 2021 at 20:31 | comment | added | edamondo | Thank you. Is there a goof reference with the proof that the dual of $L^{q}((0,T),X)$ is $L^{q'}((0,T),X')$? The concept of Bochner integral is totally new for me. | |
| Apr 22, 2021 at 18:40 | comment | added | Willie Wong | @NateEldredge since the theorem states that $X$ is reflexive, that should be enough. Probably you can just convert your comment to an answer? | |
| Apr 22, 2021 at 18:01 | comment | added | edamondo | Yes, I edited thank you | |
| Apr 22, 2021 at 18:00 | history | edited | edamondo | CC BY-SA 4.0 |
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| Apr 22, 2021 at 17:47 | comment | added | Nate Eldredge | For this to make sense, shouldn't $\phi$ take values in $X'$, not $X$? In that case, this looks like weak sequential compactness if $L^{q'}((0,T),X')$ is the dual of $L^q((0,T),X)$, which I think is true under some conditions that I forget. | |
| Apr 22, 2021 at 17:38 | comment | added | Christian Remling | I don't know if that's the case here, but it's not uncommon to use the words "compactness result" for something that just feels like some kind of compactness (some sequence has a subsequence such that ...), without necessarily being (obviously) equivalent to a statement: the subset $A$ of the topological space $\mathcal T$ is compact. | |
| Apr 22, 2021 at 17:36 | review | First posts | |||
| Apr 22, 2021 at 19:00 | |||||
| Apr 22, 2021 at 17:36 | history | asked | edamondo | CC BY-SA 4.0 |