Timeline for Regularity of Metric when defining C^k norms
Current License: CC BY-SA 4.0
3 events
| when toggle format | what | by | license | comment | |
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| Sep 26 at 16:15 | comment | added | JMK | Sure, do you have reference for that definition? I've definitely seen holder norms as $$||f||_{C^{\alpha}, g} = \sup_{x \neq y} \frac{|f(x) - f(y)|}{dist_g(x,y)^{\alpha}}$$ So I'm extrapolating from that | |
| Sep 26 at 15:55 | comment | added | mme | That is not how I would define $f'$. By contrast, if $M = \Bbb R^n$ and I assume $f(0) = 0$, you're attempting to define $f'(0) = \lim_{x\to 0} f(x)/\|x\|$. This will usually not be defined. Better is the gradient, defined by $g(\nabla f, v) = (df)(v)$, and then $\|f\|_{C^1} = \|f\|_{C^0} + \|g(\nabla f, \nabla f)\|_{C^0}$. This norm varies continuously as $g$ varies continuously in $C^0$. | |
| Sep 26 at 15:47 | history | asked | JMK | CC BY-SA 4.0 |