Timeline for Is the Banach space of continuously differential functions strongly regular?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 5, 2017 at 23:45 | comment | added | Igor Belegradek | @NateEldredge: Thank you. Your argument does work for the maximum norm $\|f\|=\max_{j,\alpha} |f^\alpha |_{C(D_j)}$ where $\alpha$ is a multi-index of order $\le k$, and $\{D_j\}$ is a finite cover of $M$ by compact disks each lying in a coordinate chart. We can assume that your sets $U_n$ lie in exactly one disk $D_j$. | |
| Mar 5, 2017 at 21:53 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| Mar 5, 2017 at 21:41 | comment | added | Nate Eldredge | Oh wait, that still needs a little work. Take $k=1$ for instance. If $|f_1|$ is small and $|\nabla f_1|$ is large, but for $f_2$ it is the other way around, then $f_1 + f_2$ might have $C^1$ norm larger than 1. | |
| Mar 5, 2017 at 21:38 | answer | added | Bill Johnson | timeline score: 3 | |
| Mar 5, 2017 at 21:35 | history | edited | Igor Belegradek | CC BY-SA 3.0 |
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| Mar 5, 2017 at 21:12 | comment | added | Nate Eldredge | For an embedded $c_0$, how about this? Choose countably many pairwise disjoint open sets $U_n \subset M$. For each one, let $f_n \in C^\infty_c(U_n)$ with $C^k$-norm one. Consider the map which sends $(a_n) \in c_0$ to $\sum a_n f_n$. | |
| Mar 5, 2017 at 21:06 | history | asked | Igor Belegradek | CC BY-SA 3.0 |