Timeline for The largest topological copy of a Hilbert space contained in $\ell^1$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 9, 2018 at 19:37 | comment | added | Pietro Majer | @LSpice of course topological embedding (not necessarily linear) need not preserve completeness- e.g. R can be embedded in R as an open interval. But the question is only meaningful in the top. linear sense ($\ell_1$ and $\ell_2$ are already homeomorphic, as topological spaces) | |
| May 9, 2018 at 19:29 | comment | added | LSpice | @PietroMajer, sorry, what I meant is: is it obvious that a mere topological embedding will preserve completeness? (Maybe the fact that the uniform, if not the metric, structure is preserved is good enough here.) | |
| May 9, 2018 at 19:03 | comment | added | ABB | @Pietro Majer: I feel these two questions will be complicated when the embedding is considered up to topological spaces. I mean the linear assumption removes. | |
| May 9, 2018 at 19:03 | comment | added | Pietro Majer | rmk: Of course if one only wants a homeo onto a subspace, not necessarily linear, then $\ell_2$ is already homeomorphic to $\ell_1$, via $\ell_2\ni x:=(x_j)_j\mapsto (|x_j|x_j)_j\in\ell_1$ | |
| May 9, 2018 at 18:57 | comment | added | Pietro Majer | @GABB In this case, note that any closed subspace is Hilbert, so we can't find $\ell_1$ | |
| May 9, 2018 at 18:55 | comment | added | Pietro Majer | @Spice yes because it is complete | |
| May 9, 2018 at 18:55 | comment | added | ABB | @Pietro Majer What happens if we change the roles? I mean what is the Hilbertian dimension of the smallest Hilbert space contains $\ell^1$ up to topological vector spaces. | |
| May 9, 2018 at 18:54 | comment | added | LSpice | Since the question is just about a topological embedding, is it obvious that the image is closed? | |
| May 9, 2018 at 18:54 | vote | accept | ABB | ||
| May 9, 2018 at 18:42 | history | answered | Pietro Majer | CC BY-SA 4.0 |