Timeline for Self-dual normed spaces which are not Hilbert spaces
Current License: CC BY-SA 3.0
11 events
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| Jan 6, 2021 at 23:48 | comment | added | M. Winter | I recently asked this question (and now came accross this post and your answer). The answer to my post also answers your questions about self-dual polytopes: there are centrally-symmetric self-dual polytopes in all dimensions. Remarkably, the answer refers to a paper in Banach space theory: Reisner, S., "Certain Banach spaces associated with graphs and CL-spaces with 1- unconditional bases". It seems this question has been asked in this context before. | |
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Feb 25, 2015 at 13:52 | comment | added | Gerald Edgar | The Denis unit ball is a square, but the unit ball for any two-dimensional Hilbert space is an ellipse. So the Denis space is not isometric to a Hilbert space. (It is, of course, linearly homeomorphic to a Hilbert space.) | |
| Feb 25, 2015 at 4:34 | comment | added | dragonxlwang | I am sorry to break the code of A&Q in this community but I don't have enough reputation to raise questions in the comment space. Would anyone care to explain to me how the example Denis constructed to be non-Hilbert? In addition, how to see that the dual space is a normed vector space? | |
| Feb 27, 2013 at 17:54 | comment | added | Mark Meckes | Unless I'm doing something silly, $\mathbb{R}^3$ equipped with the norm $\sqrt{(|x_1|+|x_2|)^2+|x_3|^2}$ is isometric to its dual. | |
| Mar 17, 2012 at 8:17 | comment | added | Denis Serre | @alvarezpaiva. This remains a question, apparently open. Even if the unit ball is a polyhedron, which has to be centro-symmetric, why should it be regular? | |
| Mar 16, 2012 at 17:39 | comment | added | alvarezpaiva | Is a self-dual norm in ${\mathbb R^3}$ necessarily Euclidean? | |
| Mar 16, 2012 at 7:36 | vote | accept | Uday | ||
| Mar 16, 2012 at 7:16 | history | edited | Denis Serre | CC BY-SA 3.0 |
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| Mar 16, 2012 at 0:55 | comment | added | Nate Eldredge | I guess the easy way to visualize this is that the unit balls in both cases are squares, but different sizes and rotated by 45 degrees. | |
| Mar 15, 2012 at 21:01 | history | answered | Denis Serre | CC BY-SA 3.0 |