Questions tagged [frechet-derivative]

The Fréchet derivative of a function from an open subset of a Banach space into another Banach space at a point is a linear map from the first Banach space into the second one which approximates particularly well the function near the given point. It generalizes the concept of derivative of a real function of one real variable.

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Graphical intuition behind general Taylor formula

I am trying to understand the visual intuition behind this formula, beyond its formal demonstration. If I think for a moment that $X$ and $Y$ are both equal to $ \mathbb{R}^2 $, I visualize this ...
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Is the inverse of an interior chart of a submanifold an immersion?

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$, $\Omega$ be an open subset of $M$, $\phi$ be a $C^1$-diffeomorphism$^1$ from $\Omega$...
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Why is $f(t) = e^{ta}$ differentiable in a unital Banach algebra?

Let $A$ be a unital Banach algebra. For $a\in A$, we define $$\exp(a):= \sum_{n=0}^\infty \frac{a^n}{n!}$$ Consider the function $$f: \Bbb{R} \to A: t \mapsto \exp(ta) = \sum_{n=0}^\infty \frac{t^n a^...
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52 views

Derivatives of mappings between normed vector spaces

I have understood that in general, given two normed spaces $(X,|\cdot |_X)$ and $(Y,|\cdot |_Y)$, an open set $U\subset X$ in $X$ and a function $f:U\to Y$, one has that: $$f':U\to \mathcal{L}(X,Y)$$ $...
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A Frechét-differentiable function $𝑓:𝑈\to 𝑌$ has a continuous derivative in $𝑥\in 𝑈\iff$ all partial derivatives are continuous in $𝑥\in 𝑈$.

In the book Zorich, Mathematical analysis II, pag. 76,77 one can read the proof of the fact that a differentiable function $f:U\to Y$ of an open subset of a normed space $X$ into a normed space $Y$ is ...
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26 views

Does Fréchet derivative needs continuity?

I want to ask a question that I feel everyone acts as if it is a common knowledge. I am really new to functional analysis. How does Fréchet differentiability imply continuity? In the definition I have ...
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How to find Frechet derivate for this functional

My functional is $F(u)=\frac{1}{2}\int (|D_x u|^2 + |D_y u|^2)dxdy + \int fu dxdy$ where $f\in C(\Omega)$, $u\in C_{0}^2(\overline{\Omega})$ and $\Omega \subset R^2$. Well Im reading Elliptic ...
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83 views

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

Let $\Lambda\subseteq\mathbb R^2$ be bounded and open, $$V:=\left\{u\in H_0^1(\Lambda,\mathbb R^d):\nabla\cdot u=0\right\}$$ and $H$ denote the completion of $V$ with respect $\left\|\;\cdot\;\right\|...
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Showing a function $f:\mathbb R^2\to \mathbb R^2$ is partially and totally differentiable

Consider $f:\mathbb R^2 \to \mathbb R, \quad (x,y) \mapsto \frac{x^3}{\sqrt{x^2+y^2}}$ and $(0,0)\mapsto 0$. How do I show that f is totally differentiable at $(0,0)$? What about showing that a ...
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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

Let $(E,d)$ be a complete separable metric space, $$|f|_{\operatorname{Lip}(d)}:=\sup_{\substack{x,\:y\:\in\:E\\x\:\ne\:y}}\frac{|f(x)-f(y)|}{d(x,y)}\;\;\;\text{for }f:E\to\mathbb R,$$ $\mu$ be a ...
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How can we show this bound for the Fréchet derivative of a Lipschitz function?

Let $E$ be a $\mathbb R$-Banach space, $v:E\to[1,\infty)$ be continuous, $$\rho(x,y):=\inf_{\substack{c\:\in\:C^1([0,\:1],\:E)\\c(0)\:=\:x\\c(1)\:=\:y}}\int_0^1v\left(c(t)\right)\left\|c'(t)\right\|_E\...
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28 views

Relation between the “modulus of gradient” to local Lipschitz continuity

Let $E$ be a $\mathbb R$-Banach space, $d$ be a metric on $E$, $f:E\to\mathbb R$, $$|f|_{\operatorname{Lip}(d)}:=\sup_{\substack{x,\:y\:\in\:E\\x\:\ne\:y}}\frac{|f(x)-f(y)|}{d(x,y)}$$ and $$|\nabla f|(...
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Can we show $\lim_{\varepsilon\to0}\sup_{0<\left\|x-y\right\|_E<\varepsilon}\frac{|f(x)-f(y)|}{\left\|x-y\right\|_E}=\left\|{\rm D}f(x)\right\|_{E'}$?

Let $f:E\to\mathbb R$ be Fréchet differentiable and $x\in E$. By definition, $$\lim_{y\to x}\frac{|f(x)-f(y)-{\rm D}f(x)(x-y)|}{\left\|x-y\right\|_E}=0.\tag1$$ Can we somehow (maybe by imposing a ...
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Are the Fréchet differentiable functions a closed subspace of this normed space of Lipschitz functions?

Let $E$ be a $\mathbb R$-Banach space, $d$ be a metric on $E$ and $\mu$ be a probability measure on $(E,\mathcal B(E))$ with $$\int d(\;\cdot\;,0)\:{\rm d}\mu<\infty\tag1.$$ Moreover, let $$|f|_{\...
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How do we norm the space of Fréchet differentiable functions from a Banach space $E$ to $\mathbb R$?

Quick question, I couldn't find an answer to: If $E$ is a $\mathbb R$-Banach space, and $C^1(E,\mathbb R)$ is the space of continuously Fréchet differentiable functions from $E$ to $\mathbb R$, how ...
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Differential operator given by the Fréchet derivative of functions over a Banach space

Let $E$ be a $\mathbb R$-Banach space, $\Omega\subseteq E$ be open and$^1$ $$\left\|f\right\|_{C^1(\Omega)}:=\left\|f\right\|_\infty+\left\|{\rm D}f\right\|_\infty\;\;\;\text{for Fréchet ...
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If $f$ is Fréchet differentiable, bound the Lipschitz seminorm by the supremum of the norms of $f$ and ${\rm D}f$

Let $E$ be a $\mathbb R$-Banach space, $$d(x,y):=\min(1,\left\|x-y\right\|_E)\;\;\;\text{for }x,y\in E,$$ $\Omega\subseteq E$ be open, $$|f|_{\operatorname{Lip}(d)}:=\sup_{\substack{x,\:y\:\in\:\Omega\...
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Uniqueness of the Frechet Derivative: the role of $x \in int_X(T)$

I'm currently trying to learn some functional analysis as a way to improve my ability to read economic theory papers. I've come across what I thought was a simple proof but on reflection I don't think ...
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Given $f$ holomorphic, which are the necessary conditions on $\phi$ in order to make $\phi \circ f \circ \phi^{-1}$ holomorphic?

It is well known that $\bar f(\bar z)$ is holomorphic whenever f is. I was wondering how to generalize this fact... Let $f: \Omega \longrightarrow \mathbb{C}$ be holomorphic and $\phi: \mathbb{C} \...
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Practical Applications of the Fréchet Derivative

I understand the definition of the Fréchet derivative. However, outside of functions on $\mathbb R^n$, I've never encountered an application where it was particularly useful. Can anyone share ...
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Derivative of homogeneous of degree p function

Let $E,F$ be Banach spaces and $U$ be an open subset of $E$ containing $0$. Suppose that $f:U\to F$ is a $C^p$ locally homogeneous of degree $p$ function. In Lang's Differnetial and Riemannian ...
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50 views

$Df = 0$ on open and connected set $\implies f$ is a constant function

Suppose $(E, \parallel \parallel),(F, \parallel \parallel)$ are Banach spaces, $U \subset E$ is an open, connected set, and $f : U \to F$ such that $Df = 0$. Prove that $f$ is a constant function. ...
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63 views

Frechet derivatives and Taylor expansion of a rational power of a matrix

Let $A$ and $B$ be two real square non-commuting matrices ($A,B \in \mathbb{R}^{N \times N}$ with $[A,B] = AB-BA \ne 0$). I also assume that $A$ is positive-definite. Consider the power function $$f : ...
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If $f$ is a $C^1$-diffeomorphism between open subsets of Banach spaces, is ${\rm D}f$ surjective at each point?

Let $E_i$ be a $\mathbb R$-Banach space, $\Omega_i\subseteq E_i$ be open and $f:\Omega_1\to\Omega_2$ be bijective. If $f$ is Fréchet differentiable at $\omega_1\in\Omega_1$ and $f^{-1}$ is Fréchet ...
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Differentiation of $l^p(\mathbb{N})$ norm

I was asked a problem, to show that $x\mapsto \Vert x\Vert_{l^p}^p=\sum_{k=0}^\infty \vert x_k\vert^p$ is differentiable at any point $x\in l^p(\mathbb{N})$ when $p\in]1,\infty[$ and twice ...
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42 views

How to show that something is Fréchet differentiable

I have that $Y$ is a complete normed linear space and denote $M=\mathcal{L}(Y)$, the space of linear operators $A:Y\rightarrow Y$. Also, let $F:M\rightarrow M$ be the map defined by $F(A)=A^2$. How ...
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56 views

calculation of Frechet derivative

Suppose we have functional $m(z,\gamma(x_{1},x_{2}))=\frac{C(z)\int\gamma(x_1,x_2)dx_1}{\gamma(x_1,x_2)}$, where $C(z)$ could be considered a constant (it doesn't change with $x_1,x_2$), $\gamma(x_1,...
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32 views

Computing the Fréchet Derivative of a Function in a Vector Space

Let $X$ be the vector space of continuous functions on the interval $[0,\pi]$ equipped with the uniform norm. Let $F: X\rightarrow X$ be a function given by $$ [F(f)](x) = \sin(f(x)),\quad (f\in X). $$...
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45 views

Total derivative of $\psi(x) = \frac{1}{\left \| x \right \|^p}Ax$

I have been trying to find the Fréchet derivative of the following function: $\psi(x) = \frac{1}{\left \| x \right \|^p}Ax$ $(x \in \mathbb{R}^n, A \in\mathbb{R^{m \times n}})$. One possibility would ...
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Fréchet derivative of $L^2\ni w\mapsto\int|w|^2$

Let $(E,\mathcal E,\lambda)$ be a measure space, $p$ be a probability density on $(E,\mathcal E,\lambda)$ and $\mu:=p\lambda$. It's easy to see that $$L^2(\mu)\ni w\mapsto\int|w|^2\:{\rm d}\mu=\left\|...
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32 views

Derivative of differential form

A smooth differential form $\omega$ on a smooth manifold $M$ is a map $\omega: M \to \bigwedge M$. Since $\bigwedge M$ is a vector bundle, and hence a manifold, if $\omega$ is differentiable as a map $...
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33 views

Smooth function of infinitely many variables

Consider $\mathbb{R}^\mathbb{N}$, the set of real-valued sequences $(x_n)_{n \in \mathbb{N}}$, as a mere set. Let $f : \mathbb{R}^\mathbb{N} \to \mathbb{R}$ be a function, and suppose that $f$ is ...
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33 views

The existence of the derivative of a Banach space valued function

In Evans' Partial Differential Equations $\S 7.1$, there is a motivation for definition of weak solution. \begin{align*} (1) \begin{cases} u_t + Lu = f \ &\text{in} \ U_T\\ u = 0 \ &\...
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A definition of differentiability that comprises Fréchet, Hadamard, and Gâteaux differentiability as special cases

Let $X,Y$ be normed spaces, $L(X,Y)$ be the set of all bounded linear operators, $\Phi$ be a set of continuous functions from $\mathbb R$ to $X$. Consider the following definition. A function $f:X \...
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77 views

Frechet Derivative simple application

Suppose that $X$ is a square integrable random variable, i.e. $X\in L^2(\Omega,\mathcal{F},\mathbb{P})$ and the function $g:L^2(\Omega,\mathcal{F},\mathbb{P})\rightarrow\mathbb{R}$ where $$g(X(\omega)...
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What is the Frechet derivative of $M_r$

Define $p_r(\hat{x})=r(\hat{x})\hat{x}, \quad \hat{x}\in \mathbb{S}^2 $ and $ r $ is scalar function on $ \mathbb{S}^2 $ $ M_r(\hat{x},\hat{y}) =\left\lbrace p_r(\hat{x})-Grad\,r(\hat{x}) \right\...
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Orthogonal derivative implies second derivative is null

Let $f:\mathbb{R}^n\to \mathbb{R}^n$ be twice differentiable, such that $f'(x)$ is an orthogonal linear transformation for every $x\in\mathbb{R}^n$. Prove that $f''(x) = 0$, for every $x\in\mathbb{R}^...
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47 views

Is the Gateaux differential an inner product if the Frechet derivative exists?

In $\mathbb{R}^n$ if the derivative of a function exists at a point $x$, then the directional derivative of that function at $x$ in the direction $v$ is an inner product between the derivative at $x$ ...
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21 views

Fréchet differentiability in Banach Spaces

If I define the operator $\varepsilon : H_{per}^{1}([0,L]) \times L_{per}^{2}([0,L]) \longrightarrow \mathbb{R}$ given by $$\varepsilon(u,v)=\int^{L}_{0} ((u_x)^2+v^2+2 \cos (u))dx,\: \forall \; u,v \...
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49 views

Does $C^1$ imply locally Lipschitz on Banach spaces?

Let $X,Y$ be Banach spaces, $\Omega \subseteq X$ open and $f: \Omega \to Y$ continuously (Fréchet-) differentiable. Does this imply that $f$ is already locally Lipschitz? For finite dimensional spaces ...
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How to get the second derivative of Fréchet of this functional?

How can I compute the second Frechet derivative of the functional $$ I(u)=\frac{1}{2}\int_{\Omega} \vert \nabla u\vert ^2\ dx \ + \dfrac{1}{p} \ \int_{\Omega}|u|^p dx$$ for $u \in H_0^{1}(\Omega)...
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1answer
163 views

Functional derivative where the functional is not an integral

Let $\phi(f)=F(f(x_1),f'(x_2),...,f^{(n)}(x_{n+1}))$ be a functional defined on $C^n([a,b])$: here $F$ is a given (say) smooth function and $x_1,...,x_{n+1}$ are given points in $[a,b]$. What is the ...
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34 views

Analytic functions having non-zero derivative preserve angle between two curves.

While studying analytic functions in complex analysis I found that if a function $f : \Bbb C \longrightarrow \Bbb C$ is an analytic function then if $\Bbb C$ is viewed as $\Bbb R^2$ then $f : \Bbb R^2 ...
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1answer
66 views

Differentiating term by term in a Banach space: how to justify it?

After looking at this question, I am now wondering if the theorem proven in the first answer below can be generalized to a Banach space. See here for my attempt. But before doing that, I have the ...
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82 views

$f(x,y) = \frac{x^3y}{x^4 + y^2}$ is not differentiable at $(0,0)$.

The directional derivative of $f: U \to \Bbb{R}^m$ at $p \in U$ in the direction $u$ is the limit, if it exists, $$\nabla_p f(u) = \lim_{t\to 0}\frac{f(p + tu) - f(p)}{t}.$$ (Often one requires that $|...
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1answer
64 views

Where is the $l_\infty$ norm Fréchet differentiable?

Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff ...
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1answer
39 views

For what $p \in [1; +\infty)$ is $l_p$ a FD-space?

Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff ...
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1answer
39 views

Is any Banach norm in $\mathbb{R}^n$ almost everywhere Fréchet differentiable?

Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff ...
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28 views

Are all FD-spaces strictly convex?

Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff ...
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48 views

Are strictly convex Banach norms Fréchet differentiable?

Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff ...

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