Timeline for Continuous automorphism groups of normed vector spaces?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2021 at 13:02 | comment | added | PseudoNeo | @BillJohnson: is it not? what's wrong with the following construction: you take a finite number of points $F$ in a very small region of the unit sphere with no (nontrivial) symmetry and take the images of this finite set by all elements of G. The convex hull $\mathbf{co}\left(\bigcup_{g\in G} gF\right)$ is the unit ball of a norm whose isometries are G, right? | |
| Feb 4, 2010 at 23:27 | comment | added | Bill Johnson | Apparently Gordon & Loewy did not know of Kalyuzhnyi's paper. Their proof is only one page (their paper is devoted mostly to other things) but the result is not obvious IMO. | |
| Feb 2, 2010 at 22:17 | comment | added | Bill Johnson | Gordon and Loewy in Math. Annalen 241, 159-180 (1979) consider the question: If $G$ is a group of linear operators on $R^n$ which contains $I$ and $-I$, is it the group of isometries of some norm on $R^n$? Among other results, they prove that the answer is yes if $G$ is finite. | |
| Feb 1, 2010 at 18:26 | vote | accept | Jason Reed | ||
| Feb 1, 2010 at 18:26 | |||||
| Feb 1, 2010 at 14:50 | vote | accept | Jason Reed | ||
| Feb 1, 2010 at 14:50 | |||||
| Feb 1, 2010 at 0:33 | comment | added | Konrad Swanepoel | @Bill: I missed your comment, which is essentially my answer. | |
| Feb 1, 2010 at 0:22 | comment | added | Bill Johnson | Thanks, Leonid (and thanks for the vote of confidence, Deane). One other comment: Any group which is the group of isometries for some n dimensional normed space $X$ must be a (necessarily compact) subgroup of the orthogonal group because isometries of it preserve the ellipsoid of maximal volume inside the unit ball of $X$. | |
| Jan 31, 2010 at 23:46 | comment | added | Deane Yang | It seems to me that someone like Bill Johnson should be given the 500 reputation points automatically. | |
| Jan 31, 2010 at 23:29 | comment | added | Bill Johnson | I see Leonid added tags, which is something I don't know how to do. | |
| Jan 31, 2010 at 23:28 | history | answered | Bill Johnson | CC BY-SA 2.5 |