Timeline for Finite imensional subspaces of $L^\infty.$
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 20, 2012 at 19:32 | comment | added | Bill Johnson | Or: every separable Banach space embeds isometrically into $C[0,1]$, which clearly embeds isometrically into $L_\infty$. | |
| Apr 20, 2012 at 19:30 | comment | added | Bill Johnson | Every separable Banach space is isometrically isomorphic to a quotient of $\ell_1$, hence every separable reflexive Banach space embeds isometrically into $\ell_\infty$, which in turn embeds isometrically into $L_\infty$. | |
| Apr 20, 2012 at 19:27 | comment | added | Igor Rivin | Maybe I did (as you see I am not very with it;) In the alluded to discussion, the OP had asked for a family of functions with prescribed $2-$ and $1-$ norms, and with sup norm bounded by $1.$ As pointed out by @fedya, this last condition is the hard one (otherwise you construct a geodesic on the sphere in three dimensions and you are done), but the set of linear combinations of three given functions with sup norm $\leq 1$ appears to be an arbitrary convex centrally symmetric set (as you seem to confirm...) so... | |
| Apr 20, 2012 at 19:24 | comment | added | Igor Rivin | That's what I thought. Why is that obvious? | |
| Apr 20, 2012 at 19:24 | comment | added | Bill Johnson | Perhaps you meant to ask something slightly different? | |
| Apr 20, 2012 at 19:23 | history | answered | Bill Johnson | CC BY-SA 3.0 |