Timeline for Norm on the space of finite signed Borel measures over a compact metric space
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 23, 2021 at 10:19 | comment | added | Robert Furber | If you want a complete norm that makes $\Delta(S)$ compact, this is impossible unless $S$ is a finite set. The usual (total variation) unit ball of $\Delta^*(S)$ will be compact with respect to this norm, and so $\Delta^*(S)$ is a countable union of compact sets. In an infinite-dimensional normed space, compact sets have empty interior, so we have a contradiction with the Baire category theorem. (This comment can become an answer if you find it satisfactory.) | |
| Apr 23, 2021 at 10:19 | comment | added | Lemma1 | Indeed, the weak-* topology will do. Thanks. | |
| Apr 22, 2021 at 14:50 | comment | added | Nate Eldredge | There won't be a norm that does it, for any practical purposes. There's a general principle that in order to construct two inequivalent complete norms on a vector space, you have to use the axiom of choice in an essential way, and so the new norm would be some horrible non-constructive monster that wouldn't likely be useful for anything. | |
| Apr 22, 2021 at 14:48 | comment | added | Nate Eldredge | The weak-* topology (induced by the duality $\Delta^*(S) = C(S)^*$) does this, right? | |
| Apr 22, 2021 at 14:09 | history | asked | Lemma1 | CC BY-SA 4.0 |