Timeline for Every linear topological space embeds into the Tychonoff product of linear metric spaces
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 18, 2019 at 18:21 | vote | accept | Taras Banakh | ||
| May 18, 2019 at 10:34 | answer | added | TaQ | timeline score: 3 | |
| May 11, 2019 at 11:51 | answer | added | user131781 | timeline score: 2 | |
| May 11, 2019 at 11:19 | history | edited | Martin Sleziak |
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| May 11, 2019 at 11:18 | comment | added | fedja | It may be there in some form in the exercise sections, but who knows? Do you have any interesting corollaries you can derive from it? And yes, you are right: I was a bit sloppy when writing the conditions on $U_k$ :-) | |
| May 11, 2019 at 11:14 | answer | added | Jochen Wengenroth | timeline score: 3 | |
| May 11, 2019 at 11:10 | comment | added | Taras Banakh | This fact should be true with a proof written by @fedja but with an extra-condition: $[-1,1]\cdot U_k\subset U_{k-1}$. Writing about "basic" fact, I did not have in mind that it is simple, but that it should be known and had to be included into basic textbooks dedicated to topological vectors spces. | |
| May 11, 2019 at 10:59 | comment | added | fedja | In other words, you claim that for every fixed neighborhood of $0$ in $X$ there is a continuous semi-metric on $X$ whose open ball is contained in that neighborhood. If I haven't made a stupid mistake somewhere, this is certainly true: just take a sequence of symmetric neighborhoods $U_k$ where $U_0$ is your original neighborhood and $(k+1)(U_{k+1}+U_{k+1})\subset U_k$ and declare the distance between $x$ and $y$ to be $\inf\{\sum_i 2^{-k_i}:x-y=\sum z_i, z_i\in U_{k_i}\}$. However (if I haven't made a mistake) it is rather a qualifier exam problem than a basic fact. Am I missing anything? | |
| May 11, 2019 at 6:36 | history | asked | Taras Banakh | CC BY-SA 4.0 |