Timeline for A question on unit norm elements of $\ell^2 \setminus \bigcup_{0<p<2 }\ell^p$
Current License: CC BY-SA 4.0
14 events
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| Aug 17 at 18:51 | comment | added | David Gao | @AliTaghavi My argument probably does not work to prove contractibility. The rotation part depends on whether the even or the odd part is not in $\ell^p$, so it’s sort of discontinuous. | |
| Aug 17 at 17:48 | comment | added | Ali Taghavi | Ok Thank you. BTW is your argument capable of being a proof of contractibility. Any way the infinite dimensional sphere is amazing: two complemented dense contractible set! | |
| Aug 17 at 17:25 | comment | added | David Gao | @AliTaghavi I believe so. The path constructed in Aleksei’s answer is rather uniform in the starting point, so you can just combine those paths into a contraction of $A$ onto a single point. There are some technical details to check, but I’m reasonably sure it should work. | |
| Aug 17 at 16:40 | comment | added | Ali Taghavi | I wondee if A is contractible too | |
| Aug 17 at 16:38 | comment | added | Ali Taghavi | no matter of cardinality I care a kind of mrasur3 or density | |
| Aug 17 at 13:53 | history | edited | David Gao | CC BY-SA 4.0 |
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| Aug 17 at 13:00 | comment | added | David Gao | @AliTaghavi I figured out a proof that $A$ is path-connected. I’ll write a proof down shortly. | |
| Aug 17 at 12:50 | comment | added | David Gao | @AliTaghavi In what sense you want to talk about being “bigger”? They have the same cardinality. They are both dense. I suspect they are both topologically speaking infinite-dimensional. And, as you already know, there’s no canonical way to define on a measure on $S$. I don’t see any other way to compare their sizes. | |
| Aug 17 at 12:36 | comment | added | David Gao | @AliTaghavi The linked paper called the unit ball “the closed solid unit sphere” and $S$ the “surface” of that. I’m aware of that, and yes, the argument in my answer works as is for $S$ being the unit sphere in the usual sense (norm exactly one). It’s just terminology differences. | |
| Aug 17 at 12:30 | comment | added | Ali Taghavi | befor I read the details, in the linked paper the set $|x|\leq 1$ is called sphere but the usual terminology is ball or disk I think. any way in my question I mean the sphere the point of unit norm not less than 1. But I guess your argument still work, yes? | |
| Aug 17 at 11:35 | comment | added | Ali Taghavi | To be honnest I doubt even A is connected but I have no idea to proof. Some how I am curious : which one is bigger A or its complement??!! | |
| Aug 17 at 11:34 | comment | added | Ali Taghavi | I am leaving for a few hours I will come back here, thanks again | |
| Aug 17 at 11:30 | comment | added | David Gao | I highly suspect $A$ is path-connected. It should be possible to “rotate” two elements of $A$ in $A$ so that their supports become disjoint, then just use a line segment (properly normalized) to connect the two. But I couldn’t figure out how to formalize this. | |
| Aug 17 at 11:27 | history | answered | David Gao | CC BY-SA 4.0 |