Timeline for Subspace of $L^2$ that lies in $L^\infty$
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Nov 1, 2022 at 14:07 | comment | added | Mohith Nagaraju | [Part 2] From this we want to conclude that $\overline{f_{1}( x)} f_{1}( x) +\cdots +\overline{f_{n}( x)} f_{n}( x) \leq C\sqrt{|f_{1}( x) |^{2} +\cdots +|f_{n}( x) |^{2}} \text{ for almost all } x.$ This might actually be completely false. For instance, it can be the case that $y\notin E_{y}$. | |
| Nov 1, 2022 at 14:07 | comment | added | Mohith Nagaraju | [Part 1] I seem to be stuck at a technical difficulty. The inequality $a_{1} f_{1}( x) \cdots +a_{n} f_{n}( x) \leq C\sqrt{|a_{1} |^{2} +\cdots +|a_{n} |^{2}}$ holds for all $x\in E_a$, where $E_a$ is some full measure set. The important point is that $E_a$ depends on $\displaystyle a_{i}$'s. Now when we put $a_{i} =\overline{f_{i}( y)}$ we have $\overline{f_{1}( y)} f_{1}( x) +\cdots +\overline{f_{n}( y)} f_{n}( x) \leq C\sqrt{|f_{1}( y) |^{2} +\cdots +|f_{n}( y) |^{2}}$. This holds for all $\displaystyle x\in E_{y}$ where $\displaystyle E_{y}$ has full measure. | |
| Jan 19, 2011 at 22:53 | comment | added | Bill Johnson | This is the "right" elementary solution and is what the makers of the qualifying exam had in mind. At a deeper conceptual level, the problem is obvious (the 2-summing norm of the identity operator on any $n$-dimensional space is $\sqrt {n}$). | |
| Jan 19, 2011 at 18:44 | vote | accept | Rostyslav Kravchenko | ||
| Jan 19, 2011 at 18:44 | comment | added | Rostyslav Kravchenko | Many thanks! This is actually the trick to extend the solution from Folland from $C[0,1]$ to $L^\infty$. | |
| Jan 19, 2011 at 17:50 | vote | accept | Rostyslav Kravchenko | ||
| Jan 19, 2011 at 17:57 | |||||
| Jan 19, 2011 at 17:06 | comment | added | Nate Eldredge | I like this!!!! | |
| Jan 19, 2011 at 16:32 | history | edited | Andrey Rekalo | CC BY-SA 2.5 |
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| Jan 19, 2011 at 16:24 | vote | accept | Rostyslav Kravchenko | ||
| Jan 19, 2011 at 16:28 | |||||
| Jan 19, 2011 at 16:08 | history | answered | Homology | CC BY-SA 2.5 |