Timeline for When a homogeneous map between vector spaces is also additive?
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 27, 2018 at 16:32 | comment | added | user111524 | I have just asked now here math.stackexchange.com/questions/2756432/… | |
| Apr 27, 2018 at 16:22 | comment | added | Joonas Ilmavirta | @users Can you ask that as a separate question? It will get much more visibility (and answers) if you post it as a question instead of a comment. I'm not sure which of MathOverflow and Math.SE is more suitable. | |
| Apr 27, 2018 at 16:04 | comment | added | user111524 | Do you have any idea about when homogeneous maps are additive if we have vector spaces over $\mathbb Q$ instead ? | |
| Dec 4, 2014 at 14:02 | comment | added | moppio89 | Ok, I will try, thank you very much! | |
| Dec 4, 2014 at 13:55 | comment | added | Joonas Ilmavirta | @moppio89, I seem to have skipped the words "or the map". I don't know what kind of a condition would be suitable if it should be simple but not immediately equivalent with additivity. Perhaps something like "the image of every convex set is convex" but I haven't checked this... | |
| Dec 4, 2014 at 13:49 | vote | accept | moppio89 | ||
| Dec 4, 2014 at 13:39 | comment | added | moppio89 | Yeah sorry, you are right, there doesn't exist any condition on the spaces which guarantee the additivity for $dim(V)>1$. I guess I woke up a little stupid this morning. . Anyhow in my question I was quite general, I wrote: "some conditions on the vector spaces or the map", what about some conditions on T? | |
| Dec 4, 2014 at 13:18 | comment | added | Joonas Ilmavirta | @moppio89, I read your question like this: "Given $V,W,T$ of which $T$ homogeneous and injective, what assumptions on $V,W$ guarantee that $T$ is additive?" A necessary and sufficient assumption is that $\dim(V)\leq1$. If $\dim(V)\geq2$, there are always nonadditive homogeneous maps so you can never deduce additivity from homogeneity. If you want to ask what conditions on $T$ guarantee additivity, that is a different question. If you want conditions on the spaces, you will have to assume that $\dim(V)\leq1$ to be able to make that conclusion. | |
| Dec 4, 2014 at 10:12 | comment | added | moppio89 | Thank you very much for your answer Joonas, but actually I asked a little different question. You showed me that an homogeneous but non additive function does exist, and I get it, but I already have a map T and I would like to know if there is some theorem which guarantees its additivity under some suitable conditions. | |
| Dec 3, 2014 at 18:05 | history | answered | Joonas Ilmavirta | CC BY-SA 3.0 |