Timeline for Norm closure of $C_b^1(\mathbb{R})$
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Feb 21, 2019 at 10:13 | comment | added | Jochen Wengenroth | I guess that you know that $BUC(\mathbb R)$ is closed in the space of bounded continuous functions endowed with the supremum norm. It thus remains to approximate every $f\in BUC$ by elements of $C_b^1$. A standard method is mollification: Fix a positive smooth function $\phi$ with compact support and integral $1$, define $\phi_n(x)=n\phi(nx)$ and consider $f\ast \phi_n$. Using $(f\ast \phi_n)'=f\ast \phi_n'$ you see that these convolutions are in $C_b^1$, and using the uniform continuity of $f$ you will show $f\ast \phi_n \to f$ uniformly. | |
| Feb 21, 2019 at 9:00 | review | Close votes | |||
| Mar 5, 2019 at 17:34 | |||||
| Feb 21, 2019 at 7:45 | review | First posts | |||
| Feb 21, 2019 at 9:48 | |||||
| Feb 21, 2019 at 7:43 | history | asked | Gergana James | CC BY-SA 4.0 |