Timeline for Prove that Cartesian composition $c_0 \times c_0$ is not isometric isomorphic [closed]
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 22, 2018 at 4:56 | history | closed |
Jan-Christoph Schlage-Puchta Yemon Choi David Handelman David Roberts♦ S. Carnahan♦ |
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| Dec 21, 2018 at 22:33 | answer | added | Dirk Werner | timeline score: 3 | |
| Dec 21, 2018 at 21:48 | comment | added | Yemon Choi | Gera, where does this problem originate? It looks to me like it could be part of an assignment | |
| Dec 21, 2018 at 18:24 | comment | added | Gera Slanova | @YemonChoi I need to find answer to the second interpretation of my question. Can you help me with this? | |
| Dec 20, 2018 at 16:29 | comment | added | Yemon Choi | I assume that by Cartesian composition you mean the direct sum of vetor spaces, and I assume that by "rate" you mean "norm". Then there are two slightly different interpretations of your question. One is to ask whether the norm you have just defined on $c_0({\bf N)}) \oplus_1 c_0({\bf N})$ is equal to the usual norm on $c_0({\bf N} \sqcup {\bf N})$, and then the answer is "no" by a trivial calculation. A slightly more interesting question (still not too difficult) is to ask whether there is any isometric isomorphism between the Banach space $c_0\oplus_1 c_0$ and $c_0$. | |
| Dec 20, 2018 at 16:26 | comment | added | Yemon Choi | @Jan-ChristophSchlage-Puchta I'm confused by your comment. $c_0$ usually denotes a specific Banach space, so that part at least is not ambiguous | |
| Dec 20, 2018 at 12:08 | comment | added | Jan-Christoph Schlage-Puchta | It seems that the answer depends on $c_0$. If $c_0=\mathbb{R}^n$, the statement is true. If $c_0=\ell^1$, it seems to be false. | |
| Dec 20, 2018 at 9:15 | review | Close votes | |||
| Dec 22, 2018 at 5:00 | |||||
| Dec 20, 2018 at 8:54 | comment | added | Fedor Petrov | what is the origin of this question and why you say "prove" (not "is it true?")? | |
| Dec 20, 2018 at 8:45 | review | First posts | |||
| Dec 20, 2018 at 8:54 | |||||
| Dec 20, 2018 at 8:40 | history | asked | Gera Slanova | CC BY-SA 4.0 |