Timeline for How to prove $C_1 \|x\|_\infty \leq \|x\| \leq C_2 \|x\|_\infty$?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 22, 2022 at 14:36 | history | edited | Calvin Khor | CC BY-SA 4.0 | fixing broken link |
| Jan 7, 2012 at 7:13 | comment | added | Rudy the Reindeer | @Fabian Nice : ) Thank you! | |
| Jan 7, 2012 at 1:42 | history | edited | J. M. ain't a mathematician | CC BY-SA 3.0 | added 15 characters in body |
| Jan 6, 2012 at 22:53 | comment | added | Fabian | @Matt: thank you for the remarks. I changed the answers accordingly. | |
| Jan 6, 2012 at 22:52 | history | edited | Fabian | CC BY-SA 3.0 | added 2 characters in body |
| Jan 5, 2012 at 14:43 | comment | added | Rudy the Reindeer | I personally find it slightly confusing to use $n$ as an index variable to sum over if $n$ also denotes the dimension of the space. | |
| Jan 5, 2012 at 14:42 | comment | added | Rudy the Reindeer | @Fabian: Assuming that $\| x \|_\infty := \max_{i \in \{1, \dots , n\} } x_i$ I think you are missing an $n$ in the following line: $$ \| x - y \| = \dots \leq n \| x - y\|_\infty$$ | |
| May 7, 2011 at 18:47 | history | edited | Fabian | CC BY-SA 3.0 | added 26 characters in body |
| Mar 5, 2011 at 18:07 | vote | accept | Huy | ||
| Mar 5, 2011 at 17:00 | comment | added | Gunnar Þór Magnússon | @Huy: If you pick a basis for your space, then you've constructed a linear isomorphism between your space and $\mathbb R^n$. These are continuous, so the Heine-Borel theorem holds on your space. Alternatively, the proof of the Heine-Borel theorem should go through mostly unchanged for a finite-dimensional normed vector space. | |
| Mar 5, 2011 at 16:37 | comment | added | Huy | This might be somewhat stupid, but is it trivial that the unit sphere is a compact set? I know Heine-Borel's theorem, but as far as I know it is only valid for subsets of $\mathbb{R}^n$. | |
| Mar 5, 2011 at 15:02 | history | answered | Fabian | CC BY-SA 2.5 |