Timeline for $C^\infty$-vectors in general representations of Lie groups on locally convex spaces
Current License: CC BY-SA 4.0
8 events
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| Sep 7, 2023 at 6:59 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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| Sep 24, 2021 at 21:56 | comment | added | paul garrett | @Z.M, I am insufficiently acquainted with Smith spaces to give a competent opinion on the history. But I can say that the assumption of quasi-complete, locally convex is sufficient to imply that closures of convex hulls of compacts are compact, and that is what gets used (in all the proofs I know) of the good behavior of vector-valued this-and-that. I had not known of any example (much less a natural one) where the compactness of closures of convex hulls of compacts would/could occur without quasi-completness (+local convexity). | |
| Sep 24, 2021 at 19:24 | comment | added | Z. M | Their $\mathcal M$-completeness seems equivalent to convex hulls of compacts being relatively compact. Let $X$ be $\mathcal M$-complete. For any compact $K\subseteq X$, pick a surjection $S\to K$ where $S$ is profinite. Then $S\to X$ extends to $\mathcal M(S)_{\le1}\to X$ of which the image is compact and contains the convex hull of $K$. On the other hand, let $X$ be a locally convex TVS where convex hulls of compacts are relatively compact, then argue as Prop 3.4 loc. cit. we deduce the completeness. Is it known that this is a filtered union of Smith spaces before Clausen-Scholze? | |
| Sep 23, 2021 at 22:24 | comment | added | paul garrett | @Z.M, ah, thanks for that fact! :) | |
| Sep 23, 2021 at 22:19 | comment | added | Z. M | This completeness does not give rise to an abelian category (cf. Question 5.1). He explained that their completeness is the possibility to integrate against any measure. It is indeed not directly related to the convergence in terms of filters if the filter is not given sequentially. | |
| Sep 23, 2021 at 21:03 | comment | added | paul garrett | It's not clear to me whether these notions are easily comparable... I'm still a little uneasy about Scholze's use of "completeness" in that and the previous section. I do suspect that some of his/their intentions are similar to the (by-now-classical) functional analysis, with one large novelty the idea to "correct" categories of TVS's to be abelian. | |
| Sep 23, 2021 at 20:23 | comment | added | Z. M | Is this completeness closely related to $\mathcal M$-completeness in Lecture 4 of Scholze's Lecture notes on Analytic Geometry? | |
| Sep 23, 2021 at 18:15 | history | answered | paul garrett | CC BY-SA 4.0 |