Timeline for Taylor expansion theorem for Gateaux differentiable functions
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 28 at 2:10 | comment | added | Deane Yang | @IosifPinelis, not exactly. I wasn’t looking for an example but perhaps a theorem where Gateaux differentiabilit is the “right” one to use. In most cases, Frechet differentiability is needed. | |
| Jan 28 at 2:05 | comment | added | Iosif Pinelis | @DeaneYang : If e.g. $f(x,y)=x^3/(x^2+y^2)$ for real $x,y$ with $x^2+y^2\ne0$ and $f(0,0)=0$, then $f$ is Gateaux-differentiable everywhere but not Fréchet-differentiable at $(0,0)$, if this is what you asked about. | |
| Jan 27 at 17:26 | comment | added | Deane Yang | Naive question: What's a situation where a Gateaux derivative is needed but a Fréchet derivative won't do? | |
| Jan 27 at 15:57 | comment | added | Alexander Schmeding | @PietroMajer : agreed . Though especially for the more exotic cases it is sometimes nice to see how this is done. For example since the Taylor theorem is a nice starting point for analytic functions (a la Bochnak and Siciaks approach to real analytic functions) | |
| Jan 27 at 11:04 | comment | added | Pietro Majer | I'd say the issue of the remainder in the Taylor expansion for a map f between Banach spaces $ X, Y$ is reduced to the case of one variable functions, that is of $t\mapsto \langle u,f(x_0+tv)\rangle\in\mathbb R$. Various assumptions (Fréchet, Gateaux, directional derivability) allows more or less uniform bounds wrto $u\in Y^*$ and $v\in X$ | |
| Jan 17 at 14:36 | comment | added | Iosif Pinelis | Do you have a response to the answers below? | |
| Jan 16 at 0:25 | comment | added | Igor Khavkine | I don't know if Gateaux derivatives are really relevant for Question 1. In the context of Banach/Hilbert manifolds, with a calculus based on the Fréchet derivative, quite adequate references are Lang's Differential and Riemannian manifolds and Dieudonné's Foundations of Modern Analysis. | |
| Jan 15 at 18:04 | history | became hot network question | |||
| Jan 15 at 15:59 | comment | added | Alexander Schmeding | Unfortunately the situation concerning the book I mentioned has not changed much. I have now a much newer improved version in front of me, but that is useless for you... | |
| Jan 15 at 15:57 | answer | added | Alexander Schmeding | timeline score: 2 | |
| Jan 15 at 14:38 | answer | added | Iosif Pinelis | timeline score: 4 | |
| S Jan 15 at 10:03 | review | First questions | |||
| Jan 15 at 13:22 | |||||
| S Jan 15 at 10:03 | history | asked | Antonio Martins Alves Veloso d | CC BY-SA 4.0 |