Can we show $\lim_{\varepsilon\to0}\sup_{0<\left\|x-y\right\|_E<\varepsilon}\frac{|f(x)-f(y)|}{\left\|x-y\right\|_E}=\left\|{\rm D}f(x)\right\|_{E'}$?

score 134 · Answer 8 · 2020-06-10 18:06:47

134

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How can we show this bound for the Fréchet derivative of a Lipschitz function?

Published on 10 Jun 2020 - 18:06

score 54 · Answer 9 · 2020-06-12 07:06:15

54

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If $d$ is a separable metric on a Banach space, approximate a $d$-Lipschitz function by Fréchet differentiable $d$-Lipschitz functions

Published on 12 Jun 2020 - 7:06

score 203 · Answer 10 · 2020-06-12 22:05:41

203

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Showing a function $f:\mathbb R^2\to \mathbb R^2$ is partially and totally differentiable

Published on 12 Jun 2020 - 22:05

score 101 · Answer 11 · 2020-06-21 04:13:00

101

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Is $H_0^1\ni u\mapsto(u\cdot\nabla)u$ Fréchet differentiable?

Published on 21 Jun 2020 - 4:13

score 30 · Answer 12 · 2020-06-30 21:59:17

30

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How to find Frechet derivate for this functional

Published on 30 Jun 2020 - 21:59

score 254 · Answer 13 · 2020-07-02 11:41:32

254

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Does Fréchet derivative needs continuity?

Published on 02 Jul 2020 - 11:41

score 737 · Answer 14 · 2020-07-04 14:26:14

737

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A Frechét-differentiable function $:\to $ has a continuous derivative in $\in \iff$ all partial derivatives are continuous in $\in $.

Published on 04 Jul 2020 - 14:26

score 182 · Answer 15 · 2020-07-05 06:31:46

182

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Derivatives of mappings between normed vector spaces

Published on 05 Jul 2020 - 6:31

List Question

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Are the Fréchet differentiable functions a closed subspace of this normed space of Lipschitz functions?

Can we show $\lim_{\varepsilon\to0}\sup_{0<\left\|x-y\right\|_E<\varepsilon}\frac{|f(x)-f(y)|}{\left\|x-y\right\|_E}=\left\|{\rm D}f(x)\right\|_{E'}$?

How can we show this bound for the Fréchet derivative of a Lipschitz function?

If $d$ is a separable metric on a Banach space, approximate a $d$-Lipschitz function by Fréchet differentiable $d$-Lipschitz functions

Showing a function $f:\mathbb R^2\to \mathbb R^2$ is partially and totally differentiable

Is $H_0^1\ni u\mapsto(u\cdot\nabla)u$ Fréchet differentiable?

How to find Frechet derivate for this functional

Does Fréchet derivative needs continuity?

A Frechét-differentiable function $:\to $ has a continuous derivative in $\in \iff$ all partial derivatives are continuous in $\in $.

Derivatives of mappings between normed vector spaces

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