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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Functional Derivative, perturbation by flow

I have read this version of the definition of functional derivative https://en.wikipedia.org/wiki/Functional_derivative#Functional_derivative In the above definition the functional derivative takes ...
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Frechet derivative of $\delta_{x(t)}$ is $\delta_{x'(t)}$?

I don't really know anything about the Frechet derivative but I was wondering if the Frechet derivative of $\delta_{x(t)}$ was $\delta_{x'(t)}$. More precisely, if we consider the banach space $(\...
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Fréchet derivative of inner product of <x,Ay> [closed]

Quick and maybe easy question. Is the Fréchet derivative of this inner product correct: \begin{align*} \frac{d}{dA}\left\langle x,Ay\right\rangle & =x\otimes y \end{align*} $x,y\in X$ belong to ...
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How this functional differentiable?

Here is a a given functional $$G(u)=\int_{S^1}u^{p}(x)dx-1,$$ where $u\in H^1(S^1).$ I wonder why it is continuously Fréchet differentiable in $H^1(S^1)$ with $$DG(u)=p\int_{S^1}u^{p-1}dx,$$ when $p&...
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Finding the Frechet derivative of $f:\mathbb{R}^{N\times N} \to \mathbb{R}^{N\times N}, X\mapsto X^2$

We just introduced the concept of the Frechet-derivative in class and I'm trying to grasp how to do it so I want to know wether my understanding is correct. Let's consider $$f:\mathbb{R}^{N\times N} \...
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Equivalent definitions of Fréchet differentiability

In their book The Ricci Flow in Riemannian Geometry Andrews and Hopper have an appendix on Gâteaux and Fréchet differentiability. They define Gâteaux differentiability as follows: Let $X,Y$ be Banach ...
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Generalizing complex derivative as Fréchet/Gateaux derivative

So it is well-known that complex differentiability of a function $f:\mathbb{C}\rightarrow\mathbb{C}$ is equivalent to the function being Fréchet/Gateaux differentiable and the component functions (...
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Proof that, if the norm of any normed space is (Fréchet) differentiable, then the derivative is continuous.

I want to prove the following lemma. Let $X$ be a normed space and its norm $\|.\|$ be Fréchet differentiable on $X \backslash \{0\}$. Then the derivative is continuous there. I've got this from a ...
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In which points is the supremum norm on $C[0,1]$ and $c_0$, respectively, Gâteaux/Fréchet differentiable?

$f : U \to Y$ where $U\subset X$ is open and $X, Y$ are normed spaces is called Gâteaux differetiable at $u\in U$ if there exists a bounded linear operator $T$ from $U$ to $Y$ such that for $h\to 0$ ...
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If a function is separately differentiable is it diferentiable?

Let $X $, $Y $ and $Z $ normed vector spaces $$f:X \times Y \to Z$$ Such that $f(x, \cdot)$ is differentiable for all $x \in X $ and $f (\cdot ,y)$ is differentiable for all $y \in Y $. Is $f $ ...
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Is the sine operator on $L^2[0,1]$ Fréchet differentiable or not and why?

This problem has given me some trouble. Let $F$ be the operator on $L^2[0,1]$ defined by $F(g)(t)=\sin g(t)$. I'm trying to determine whether or not $F$ is (Fréchet) differentiable in that space. I ...
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Range preserving matrix derivative

Let $M(x) \in \mathbb{R}^{d\times d}$ be a matrix-valued function on $\mathbb{R}^d$ and let $F(x) = M(x)x$. Suppose $M(x)$ is differentiable at $x$ and the rank of $M$ is constant for an openset $\...
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Existence of Frechet Derivative of Log-Normalizer for Exponential Family Type Distribution

Let's say we have a density function that generalizes the exponential family slightly \begin{align} p(t)=q(t)\exp(f(t)-A(f)) \end{align} where if $f(t)=\theta^T \phi(t)$ we recover the exponential ...
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Relation between Frechet derivative and convex functions

I am studying Fréchet derivatives and I am trying to show that, with we let $(V,\|\cdot\|_V)$ be a normed vector space and $U\subset V$ an open and convex subset, if we define $f:U\rightarrow \mathbb{...
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Fréchet derivatives for $f: \mathbb{R}^{d\times d} \rightarrow \mathbb{R}^{d\times d}_{sim}$

I am starting my studies at Fréchet derivatives and I saw the exercise below: Let $\mathbb{R}^{d \times d}$ with the usual operator norm. We know it is a Banach space. Let $\mathbb{R}^{d\times d}_{...
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Example of function that is Gâteaux-differentiable but not Fréchet-differentiable

I am looking for an example of a function that is Gateaux-differentiable but not Fréchet-differentiable. I know that there is a lot of example of function $f: \mathbb R^2 \to \mathbb R$ that satisfies ...
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Sard's Theorem with integration

I want to prove this: Let $U\subset\mathbb{R}^n$ be an open set, $f \in \mathcal{C}^1(U;\mathbb{R}^n)$ and $E\subset U$ measurable. Prove that $f(E)$ is measurable and $$\mu (f(E)) \leq \displaystyle \...
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Fréchet derivative of a (nonlinear) differential operator

The question may not be too well-posed, but loosely speaking, suppose $L:W^{1,p}(\mathbb R)\to L^p(\mathbb R)$ is a (possibly nonlinear) first order differential differential operator, such that all ...
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If $f \in W^{1, 1}(\mathbb{R}^n)$ then $\theta \mapsto f \circ (Id + \theta)$ is differentiable

In a book I'm reading I have this statement: If $f \in W^{1, 1}(\mathbb{R}^N)$, then the map defined by \begin{align} W^{1, \infty}(\mathbb{R}^N; \mathbb{R}^N) &\rightarrow L^1(\mathbb{R}^N)\\ \...
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Chain rule involving Fréchet derivative

Suppose that $P(w)$ is a probability density function with support $w \in [0,\infty)$ and $G = G[P]$ is a functional satisfying $G[P] \in [0,1]$. I saw a paper used a chain rule of the following form: ...
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Leibniz rule for vector-valued functions

If $f \in C^\infty(\mathbb{R}^d, \mathbb{R})$ and $g \in C^\infty(\mathbb{R}^d,F)$ for some Fréchet space $F$, what are the derivatives of their pointwise product $fg$? I guess that it has to be $$D^n(...
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Is there a Fréchet Derivative for the Norm on a Complex Hilbert Space?

Let $\mathcal H$ be a complex Hilbert space and consider the function $\langle\cdot,\cdot \rangle = \|\cdot\|^2:\mathcal H \to \mathbb R \subset \mathbb C$, which is the square of the norm induced by ...
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How to understand this equation $L(L^2(a,b), \mathbb{R}) = L^2(a,b)$?

In our lecture there is the following sentence: "The function $J:H^1_{a}(a,b) \times L^\infty(a,b) \to \mathbb{R}$ is assumed to be Fréchet differentiable. We assume the partial derivatives can ...
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Find the $(n +1)$-th derivative of the function $f_A.$

Let $E,F$ be Banach spaces and $A \in \mathcal L (\underbrace {E,\cdots, E}_{n\ \text {times}}\ ; F)$ i.e. $A$ is a bounded multilinear map. Compute the successive derivatives of the mapping $$f_A : E ...
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Fréchet derivative of scalarfield operator $ f \mapsto \nabla f \cdot \sigma $?

Let $\sigma \in \mathfrak{X}(\Omega)$ be a fixed smooth vector field on an open domain $\Omega \subset \mathbb{R}^d$. Consider the operator $T\colon C^\infty(\Omega) \to C^\infty(\Omega)$ on the space ...
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Determine the tangent space of the boundary of a manifold of the form $f(\mathbb R^d)$ for a differentiable $f$

Let $d\in\mathbb N$; $x^\ast\in\mathbb R^d$; $k\in\mathbb N$; $f:\mathbb R^d\to\mathbb R^k$ be Fréchet differentiable at $x^\ast$; $v^\ast:=f(x^\ast)$; $A:={\rm D}f(x)$; $M:=f(\mathbb R^d)$. Assume ...
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Best linear approximation in a neighborhood property for Frechet derivative

Let $D$ denote a Frechet derivative at a point $a$. Then for $f:\mathbb{R}\to\mathbb{R}$ it holds for all $x$ from some neighborhood of $a$: $$|f(x)-(f(a)+D(x-a))|\leq|f(x)-(f(a)+M(x-a))|,\ M\neq D$$ ...
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66 views

what is the difference between Strong Fréchet derivative and Fréchet derivative

I am confused about the relationship between Frechet differentiable and strong Fréchet differentiable. Assume the function $f(x) \in \mathbb{R}, x\in \mathbb{R}^n$ is Fréchet differentiable at $x$. ...
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If $f$ is an isometry between open sets, is $\left|\det{\rm D}f\right|=1$?

Let $d\in\mathbb N$ and $\Omega_i\subseteq\mathbb R^d$ be open. If $f$ is an isometry from $\Omega_1$ to $\Omega_2$, can we conclude that $f$ is Fréchet differentiable and $\left|\det{\rm D}f\right|=1$...
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Difficulty in proving that $\Phi$ is differentiable.

I am studying Lie algebra and Lie groups from the lecture notes given by our instructor. Here I find a definition of differentiable Lie algebra homomorphism. What it says is as follows $:$ Let $E$ be ...
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How to prove that $\phi$ is continuous?

Definition $:$ Let $E$ be a Banach space. Let $G,H \subseteq GL(E)$ be linear Banach Lie groups (i.e. they are closed subgroups of $GL (E)$). A group homomorphism $\Phi : G \longrightarrow H$ is said ...
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Local integrability of a Lebesgue integral

Fix a Banach space $X$, an open subset $U \subset X$, and a measure space $(\Omega, \mathcal A, \mu)$. Let $f: U \times \Omega \to \mathcal R$ be continuously Frechet-differentiable in the first ...
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54 views

Prove that $X \in \mathfrak g.$

Let $E$ be a Banach space. Let $G \subseteq GL(E)$ be a closed subgroup i.e. $G$ is a linear Banach Lie group. Define $$\mathfrak g : = \left \{X \in \mathcal L(E)\ \big |\ \forall t \in \Bbb R, e^{tX}...
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Do you know any result on how to interchange stochastic integration with Frechet derivative?

Let we have a stochastic basis $(\Omega,\mathcal{F},\mathbb{F} = (\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ with standard assumptions. Let the integral $\int_0^t f(s,h) dB_s$, for $f(s,h)$ a ...
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How to show that $D_{f}$ is a Borel function

How to show that $D_{f}$ is a Borel function. Well I have one Lipschitz function $f:\Bbb{R}^{n}\to \Bbb{R}$ and I want to proof that $D_{f}:D\to L(\Bbb{R}^{n},\Bbb{R})$ is Borel function, where $D=\{ ...
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How to show frechet characterization differentiability 2

Let $U\subset R^{n} $ and $f:U\to R$. Show that: $f$ is differetiable in $x_0\in U$ iff exist $A\in L(R^{n},R)$ and exist $r>0$: $\lim_{t\rightarrow 0^{+}}\frac{f(x_0 +tv)-f(x_0)}{t}=Av$ uniformly ...
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Fréchet derivative of $\|Au-f\|^2$

$A$ is a bounded linear operator on an infinite dimensional Hilbert space. I have been told the answer is supposed to be $2A^{*}(Au-f)$. Here is my progress: $\|A(u+h)-f\|^2-\|Au-f\|^2=\langle A(u+h)-...
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41 views

Prove that $\ddot {g} (t) = D^2 f(x+th) (h,h).$

Let $E$ be a Banach space. Let $U \subseteq E$ be open and $f : U \longrightarrow F$ be $C^2.$ Let $x \in U,h \in E$ be such that $B(x, \|h\|) \subseteq U.$ Consider the map $g : (-1,1) \...
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Prove that $\dot {\gamma} (t)$ is a skew symmetric matrix.

Let $\gamma : (-1,1) \longrightarrow \Bbb M_n(\Bbb R)$ be a $C^1$ map such that $\gamma (t) \in O(n),\ \forall t.$ Then show that for all $t,$ $\dot {\gamma} (t)$ is a skew-symmetric matrix. My ...
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Example of calculation of Frechet derivative considering Hilbert space

Let $\mathbb R$ a Hilbert space in their canonic form. We define the function $f(x)=a\|x-x^{o}\|^{2}+\langle b,x \rangle $, $\forall x \in \mathbb R^{n}$, where $a$ is a constant, while $x^{o}$ and $b$...
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36 views

Prove that $\dot {\gamma_w} (0) = w.$

Let $\Bbb E^n$ be the Euclidean $n$-space. Let $p \in \Bbb S^{n-1},$ the $(n-1)$-sphere in $\Bbb E^n.$ Let $w \in T_p \Bbb S^{n-1},$ the subspace of $\Bbb E^n$ parallel to the affine subspace tangent ...
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181 views

Understanding Leibniz rule in the framework of Banach space.

Let $E,F_1,F_2,G$ be Banach spaces and $U \subseteq E$ be an open subset. Let $f_j : U \longrightarrow F_j$ be $C^r$ maps and $B : F_1 \times F_2 \longrightarrow G$ be a bounded bilinear map. Then $g :...
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Understanding the proof of the derivative of a multi-linear map.

Let $(E_1, \| \cdot \|_{E_1}),(E_2, \|\cdot\|), \cdots, (E_n,\|\cdot\|_{E_n})$ be Banach spaces. Let $\displaystyle E = \mathop{\oplus}_{i=1}^{n} E_i.$ Define a norm $\|\cdot\|_{\infty}$ on $E$ by $$\|...
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35 views

Boundedness of the first Fréchet derivative

Let $(E,\Vert\cdot\Vert)$ be a normed vector space and $f:E \rightarrow \mathbb{R}$ be a twice Fréchet differentiable function with $\sup_{x \in E} \frac{\vert f(x)\vert}{1+\Vert x\Vert^3} < \infty$...
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Two versions of Fréchet derivatives in terms of Landau notation

In the wiki article, $F$ is Fréchet differentiable at $x$ if there exists a function $A$ that is linear in $h$ such that $F(x+h)-F(x)-Ah=R(h)$ is $o(h)$, meaning that for every $\epsilon> 0$, there ...
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41 views

Does Frechet differentiability of the first order partials imply twice Frechet differentiability?

Let $f:\mathbb{R}^n \to \mathbb{R}$. If all first order partial derivatives of $f$ are Frechet differentiable, must $f$ be twice Frechet differentiable? This question is inspired by my comment on this ...
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62 views

Fréchet Derivative of $\frac{1}{\|x\|}$

In a Hilbert Space $V$, for function $f:V\to\mathbb{R}$, if $f$ is Fréchet differentiable at $x_0$, the Fréchet Derivative $\nabla f(x_0)$ is $v$ such that $$ \lim_{x\to x_0} \frac{|f(x) - f(x_0) - \...
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57 views

Derivative of a Functional. The Chain Rule? Whats going on here.

Hi suppose I have a functional $I_{x,y}$ acting on continuous paths $$\{\xi~:~\xi:[0,1]\to \mathbb{R}^n,~\xi(0)=x,~\xi(1)=y\}$$ . In my case $I_{x,y}[\xi]=\int_0^1 L(\xi(t),\dot{\xi}(t))dt$ where $L(\...
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1answer
46 views

Frechet derivative of trace of matrix expression

I am new to the concept of Frechet Derivatives. I have encountered a problem where I am supposed to find the Frechet derivative of $\operatorname{trace}(XAX+AXA^T)$ where my $X,A \in \Bbb R^{n\times n}...
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41 views

How to compute Frechet Derivatives that involve matrices.

Suppose I have a linear operator which maps $S^n\to\mathbb R^{n \times n}$. It is represented as $LX:= XA^T + AX.$ How can I find the Frechet derivative of $f(X) = XA^T + AX.$

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