Timeline for Subspaces of finite codimension in Banach spaces
Current License: CC BY-SA 2.5
3 events
| when toggle format | what | by | license | comment | |
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| Jul 7, 2010 at 12:35 | comment | added | Keivan Karai | you are right. This is a functional which is not of inner product with any element in the pre-Hilbert space (C[-1,1] considered with the $l^2$ norm). | |
| Jul 7, 2010 at 11:18 | comment | added | Matthew Daws | I don't agree: pick some function g with T(g)=1. (g(x)=x+1 works I think). Then define a map $P:C[-1,1]\rightarrow C[-1,1]$ by $P(f) = f - T(f)g$. If $T(f)=0$ then $P(f)=f$, while also $TP(f) = T(f) - T(f)T(g) = 0$ for all $f$. So $P$ is a projection onto the kernel. It seems like you are trying to treat $C[-1,1]$ as an inner-product space (dense in $L^2[-1,1]$ for example) and then trying to pick an orthogonal projection: indeed, this does not exist. | |
| Jul 7, 2010 at 11:05 | history | answered | Keivan Karai | CC BY-SA 2.5 |