Timeline for Faa di Bruno's formula for vector valued functions
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 3 at 14:17 | comment | added | T. Amdeberhan | @JochenWengenroth: can you elaborate how you use Hahn-Banach here? | |
| Sep 8, 2012 at 18:14 | vote | accept | M-S | ||
| Sep 5, 2012 at 8:24 | comment | added | Jochen Wengenroth | Elaborating Gerald's comment: A way to verify a formula in a Banach space is to use the Hahn-Banach theorem: $x=y$ holds if and only if $\phi(x)=\phi(y)$ for all continuous linear functionals $\phi$. This leads to a reduction to a scalar case and you do not even need to know the scalar proof. | |
| Sep 4, 2012 at 18:33 | comment | added | Gerald Edgar | It seems to be copied wrong. But yes, once you get the formula for scalar values, the same one holds when the outer function has vector values. But the thing to do is provide the proof. Use the scalar theorem to prove the vector theorem. | |
| Sep 4, 2012 at 16:27 | comment | added | M-S | In Schwartz, Analysis, vol.I, is the following formula for scalar functions: $$ (f\circ g)^{(n)}(x)=\sum_{k_1+k_2+...+k_m=m} \frac{m!}{k_1! k_2! \ldots k_m! (1!)^{k_1} (2!)^{k_2} \ldots (m!)^{k_m} } $$ $$ g^{(k_1+k_2+\ldots +k_m)}(f(a)) (f')^{k_1}(a) (f'')^{k_2}(a) \ldots (f^{(m)})^{k_m}(a). $$ (the same is in Wikipedia) Is this formula true if $f,g$ are as in my question? | |
| Sep 4, 2012 at 15:20 | comment | added | Gerald Edgar | MUCH more complicated than the question asked! | |
| Sep 4, 2012 at 12:48 | history | answered | Bazin | CC BY-SA 3.0 |