Timeline for Contractions and spaces
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
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| Jun 12, 2010 at 15:35 | comment | added | Bill Johnson | @ Ady, the theorem is that any separable space can be equivalently renormed so that plus and minus the identity are the only isometries, and I think that is for into isometries. Sorry I botched the comment above and that I don't recall the history or who proved the final theorem, but that is in some other MO thread... | |
| Jun 12, 2010 at 5:12 | comment | added | Ady | @ Professor Johnson. Thank you, but I do not understand the answer. Since I was speaking about into isometries. Also, I said "having not GL-l.u.st." | |
| Jun 12, 2010 at 2:53 | comment | added | Bill Johnson | @ Ady: I don't know about the first question. The answer to the second question is "no"-- $\ell_2$ has an equivalent renorming so that the only isometries are plus and minus the identity. This was proved by W. J. Davis IIRC (and was discussed on another MO thread where I got the history wrong and now am not sure of the correct history). | |
| Jun 11, 2010 at 11:36 | comment | added | Ady | @ Professor Johnson. Just for the sake of curiosity. Is the same thing true for the family of their "cousins", namely the Sobolev spaces $W^{k,1}(\mathbb{R}^{n})$ with $k=1,2,...$ and $n\geq2$ , which are not $\mathcal{L}_{1}$- spaces ? Or, more generally, is it true that every Banach space having not GL-l.u.st. contains an isometric, non-complemented copy of itself ? | |
| Jun 10, 2010 at 8:46 | history | undeleted | Bill Johnson | ||
| Jun 10, 2010 at 7:21 | history | deleted | Bill Johnson | ||
| Jun 10, 2010 at 7:20 | history | answered | Bill Johnson | CC BY-SA 2.5 |