Timeline for Completeness of a normed space
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 2, 2023 at 12:25 | comment | added | Mathlover | Thank you sir for your help. | |
| Jun 30, 2023 at 17:04 | comment | added | Pietro Majer | Of course this space can't be complete . Think of $X=\mathbb R$, $g:=e^\theta$ (if you want an example with $\rho<\infty$, or just $g:=1$ to make it simpler). Then $B_g$ is dense in $L^1$, where you can find functions that are everywhere discontinuous and unbounded. | |
| Jun 28, 2023 at 20:33 | comment | added | Mathlover | Thank you for your response. Could you please provide more explicit details or clarification? | |
| Jun 28, 2023 at 20:17 | comment | added | Willie Wong | I find it hard to believe that $\mathcal{B}_g$ is complete. The inverse of the Cantor function (take a caglad version) can be expressed as the uniform limit of caglad functions , each of which with finitely many discontinuities. Under you hypotheses, uniform convergence implies $\mathcal{B}_g$ convergence, but the inverse Cantor function is not in $\mathcal{B}_g$ as you defined it, as it has infinitely many discontinuity points. | |
| Jun 28, 2023 at 19:45 | history | asked | Mathlover | CC BY-SA 4.0 |