Timeline for $C[0,1]$ is Banach-space isomorphic to $c_0(C[0,1])$
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 26, 2016 at 4:26 | comment | added | Bill Johnson | Fedor, you don't need Milutin's theorem. To see that $C[0,1]$ is isomorphic to $C[0,1]$, consider the subspaces of functions that vanish on $[1/2,1]$ and vanish on $[0,1/2]$ to see that the $\infty$-direct sum of two hyperplanes in $C[0,1]$ is (even isometrically) isomorphic to a hyperplane in $C[0,1]$. The space $C[0,1]$ is isomorphic to its hyperplanes because, for example, it contains a complemented subspace isomorphic to $c_0$. | |
| Feb 25, 2016 at 14:45 | comment | added | Li Jingyang | Let us continue this discussion in chat. | |
| Feb 25, 2016 at 14:42 | comment | added | Fedor Petrov | What may be easier? | |
| Feb 25, 2016 at 14:15 | vote | accept | Li Jingyang | ||
| Feb 25, 2016 at 14:14 | comment | added | Li Jingyang | oh, I see. thank you. $Y=\{f\in C(K)| f(0,0)=0\}$. I can chek this statemet by building an isometric isomorphism. But is there any easier way to see this? | |
| Feb 25, 2016 at 14:10 | history | edited | Fedor Petrov | CC BY-SA 3.0 |
added 20 characters in body
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| Feb 25, 2016 at 14:10 | comment | added | Fedor Petrov | No, it is another assumption. Sorry, I forgot to formulate the conclusion, fixed now. | |
| Feb 25, 2016 at 14:04 | comment | added | Li Jingyang | @FedorPetrov.Thank you for help. I am confused at the statement of your lemma. Do you mean that if $X\sim X\oplus X, Y\sim \oplus Y$, "Then" each of $X,Y$ is isomorphic to a complemented subspace of another? | |
| Feb 25, 2016 at 13:50 | comment | added | Fedor Petrov | Complement subspace. | |
| Feb 25, 2016 at 13:40 | comment | added | Li Jingyang | @FedorPetrov.What is "A" in the proof of the lemma above? | |
| Feb 25, 2016 at 13:21 | history | answered | Fedor Petrov | CC BY-SA 3.0 |