Timeline for Is this set of functions compact?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 29, 2011 at 16:21 | comment | added | NTT | @RW with respect to both \varphi and $\delta$. | |
| Aug 26, 2011 at 20:38 | comment | added | R W | "Uniformly bounded and continuous" with respect to which parameters? Both $\varphi$ and $\delta$ or just one of them (which?) - provided the other one is fixed? | |
| Aug 26, 2011 at 2:19 | comment | added | NTT | @RW, if I can prove that the $\varphi_\delta$s are uniformly bounded and equicontinuous, would there be a known result that can help me reach compactness? | |
| Aug 25, 2011 at 7:46 | comment | added | R W | It seems that such an argument (if it exists) would still be equivalent to proving triviality of $\mathcal F$ (or a little bit weaker). The point is that if you look just at bounded harmonic functions (be it in your setup or for more general Riemannian manifolds or Markov operators), then they also form a Banach space (with respect to the sup norm), and its unit ball is compact only if the space is finite dimensional. | |
| Aug 24, 2011 at 21:15 | comment | added | NTT | @RW, yes! I'm asking for an argument without prior knowledge that $\mathcal{F}$ is trivial. | |
| Aug 24, 2011 at 16:56 | comment | added | R W | Since you define the functions $\phi_\delta$ as averages of functions $\phi$ from the set $\mathcal F$ which consists of constants only, all $\phi_\delta$ are also constants, aren't they? Or are you asking for an argument working without prior knowledge that $\mathcal F$ is trivial? | |
| Aug 23, 2011 at 18:21 | comment | added | NTT | Thanks for pointing this out! I still would like to have an answer for my original question (i.e. compactness of the set defined above). | |
| Aug 23, 2011 at 4:53 | history | answered | R W | CC BY-SA 3.0 |