Timeline for Do (Banach) ultrapowers carry some sort of 'elementary equivalence'?
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 14, 2013 at 13:47 | comment | added | Tomasz Kania | Let me point out that ultraproducts of Banach spaces are never injective (unless finite-dimensional); see matematicas.unex.es/~fcabello/files/printable/66sub.pdf Theorem 6.1 | |
| Apr 20, 2012 at 15:34 | comment | added | Bill Johnson | ...the sum of any two totally incomparable closed subspaces of any Banach space is closed. | |
| Apr 20, 2012 at 15:33 | comment | added | Bill Johnson | If $X$ is a subspace of $\ell_2 \oplus c_0$ that is isomorphic to $\ell_2$, then, since the projection onto $c_0$ is strictly singular when restricted to $X$, the projection onto $\ell_2$ is an isomorphism on some finite codimensional subspace of $X$. On the other hand, if $Y$ is subspace of $\ell_2 \oplus c_0$ that is isomorphic to $c_0$, then the projection onto $\ell_2$ is compact on $Y$. Play with these two facts to see that $X + Y$ is complemented (note that it is automatically closed by the two facts since e.g. the intersection of $X$ and $Y$ must be finite dimensional; in fact,...). | |
| Apr 20, 2012 at 11:27 | comment | added | Olaf Kummers | A wonderful answer. Unfortunately, the above-mentioned property for $\ell_2\oplus c_0$ is not clear to me... | |
| Apr 19, 2012 at 17:59 | vote | accept | Olaf Kummers | ||
| Apr 18, 2012 at 18:24 | history | edited | Bill Johnson | CC BY-SA 3.0 |
Corrected typo.
|
| Apr 18, 2012 at 15:10 | history | answered | Bill Johnson | CC BY-SA 3.0 |