Timeline for Tensor product is complete?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 24, 2022 at 10:51 | comment | added | Martin Geller | Yes, the symmetry requirement only works for copies of the same $V$ tensored with each other. This is an oversight | |
| Aug 23, 2022 at 21:55 | comment | added | KConrad | I have rereread the question multiple times and there is something I don't get: what could it mean to say $||v \otimes w||_{V \otimes W} = ||w \otimes v||$ when $V$ and $W$ are not the same space? The elementary tensor $w \otimes v$ doesn't live in $V \otimes W$, so the right side of that equation doesn't make sense to me. | |
| Aug 23, 2022 at 19:29 | comment | added | Nik Weaver | @terceira it sure doesn't look like it could be in the algebraic tensor product, but can you prove this? | |
| Aug 23, 2022 at 18:41 | comment | added | terceira | A suggestion to show that the result is false if both spaces are infinite dimensional. One can find norm one linearly independent sequences $(x_n)$ and $(y_n)$. The series $\Sigma \frac 1 {n^2}x_n \otimes y_n$ converges in the completion but the limit is not in the algebraic tensor product. | |
| Aug 23, 2022 at 16:21 | comment | added | Nik Weaver | Good idea ... is the set of rank $\leq r$ tensors closed? I guess this would do it. | |
| Aug 23, 2022 at 15:55 | review | Close votes | |||
| Aug 28, 2022 at 3:01 | |||||
| Aug 23, 2022 at 15:51 | answer | added | Gerald Edgar | timeline score: 1 | |
| Aug 23, 2022 at 14:38 | comment | added | Nik Weaver | In fact, it's hard to think of any examples where the algebraic tensor product $V\otimes W$ is complete. Conjecture: if $V$ and $W$ are both infinite dimensional, then $V\otimes W$ is not complete. | |
| Aug 23, 2022 at 14:31 | comment | added | Onur Oktay | Let's assume both V and W are infinite dimensional. Injective tensor product and projective tensor product are the very first examples that comes to mind for which $V\otimes W$ is not complete. | |
| Aug 23, 2022 at 14:15 | history | asked | Martin Geller | CC BY-SA 4.0 |