Timeline for Isometries between subspaces of finite-dimensional vector spaces
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| S Oct 2, 2018 at 18:38 | history | suggested | user127987 | CC BY-SA 4.0 |
corrected notation
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| Oct 2, 2018 at 14:19 | review | Suggested edits | |||
| S Oct 2, 2018 at 18:38 | |||||
| Jan 23, 2018 at 23:53 | comment | added | Beata Randrianantoanina | If you mean how different can the subspaces be that isometric to each other, then I recommend that you look at my paper available on Arxiv math.FA/9709214, where I show an example of two subspaces of L_p, where p\ne 2 is an even integer, so that they are isometric to each other and one subspace is complemented in L_p, while the other one is not complemented. You should check if and how this can be adapted for the finite dimensional case. | |
| Jan 23, 2018 at 20:05 | comment | added | Dave | Unfortunately I cannot edit my original question, but I would like to know which subspaces of $\ell_p^n$ are isometric, for $p$ an even integer. My second paragraph describes a special case of the result you just mentioned. To explain: $\ell_p^n$ can be identified as a subspace of $\ell_p^m$, and the subspaces isometric to this subspace are given as the range of $A$, where $A$ is described in my question. What about subspaces that are not spanned by vectors with disjoint support? We can characterise all such isometries for any $p$ that is not an even integer. What is known for even $p$? | |
| Jan 23, 2018 at 18:54 | comment | added | Beata Randrianantoanina | I am still not sure what you mean exactly, especially in your second paragraph. Or what do you want your subspaces to be isometric to? A lot is known about isometries of $\ell_p$ and $\ell_p^n$. In particular any subspace spanned by disjointly supported vectors is isometric to $\ell_p$ of appropriate dimension. | |
| Jan 20, 2018 at 14:08 | comment | added | Dave | Apologies for my mistake; I mean $\ell_p^n(\mathbb{R})$ instead of $\ell_p^n(\mathbb{R}^n)$. The $n$-dimensional real vector space equipped with the $\ell_p$-norm. | |
| Jan 20, 2018 at 6:20 | comment | added | Beata Randrianantoanina | I am somewhat confused by your notation. What do you mean by $\mathbb{R}^n$? Do you mean the $\ell_2$-norm on it? It would be nice if you could fix the various typos in your question. About a reference, I am not sure if this is what you are looking for but in one of my old papers I considered surjective isometries of related spaces, possibly you can deduce something for your situation from it. The paper is available at arxiv.org/pdf/math/9604216.pdf. | |
| Jan 18, 2018 at 23:18 | review | First posts | |||
| Jan 19, 2018 at 1:45 | |||||
| Jan 18, 2018 at 23:17 | history | asked | Dave | CC BY-SA 3.0 |