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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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Fréchet Differentiability of a series of norms

I am currently trying to understand the proof of Theorem 9.14 from "Banach Space Theory",Fabian M. et al, which characterizes super-reflexivity. The theorem affirms the following: Let $X$ be ...
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Uniform Fréchet Differentiability of a series of norms

I'm studying Theorem 9.14 from "Banach Space Theory",Fabian M. et al, which characterizes super-reflexivity. One of the steps of the proof is to show that 2 and 3 imply 4. So, using two ...
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Uniform Fréchet differentiability

Right now, I'm studying concepts of differentiation in Banach spaces, but I'm pretty new. In several references, I've found the following property: "Let $U\subset X$ be an open convex subset of a ...
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Does the Fréchet derivative of a function $f: X \to \mathbb R$ vanish at an extremum?

Let $X$ be a Banach space with norm $\lVert \cdot\rVert _X$ and $f: X \to \mathbb R$ a Fréchet-differentiable function, so there exists $f': X \to X^*$ such that $$ \lim_{h \to 0} \frac{f(x+h)-f(x)-f'(...
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Higher order Frechet derivatives viewed as bilinear maps, on Taylors theorem

So I have been studying some introductory non-linear analysis. I am currently looking at higher order Frechet derivatives and I want to proof-check/ make sure I got something right. So given $X,Y$ ...
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why we are looking for a bounded linear operator in definition of Frechet differentiability?

maybe this is a silly question to ask but I was wondering why in the definition of Frechet differentiability of a map between X and Y, we are saying if there exist a bounded linear operators such that ...
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Consistency of bootstrap estimator that is continuously $\rho_\infty$-Frechet differentiable

Theorem. Suppose $T$ is continuously $\rho_r$-Frechet differentiable at $F$ with the influence function satisfying $0<E[\phi_F(X_1)]^2<\infty$ and that $\int F(x)[1-F(x)]^{r/2}dx<\infty$. ...
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Matrix function derivative. Introduction

The author of this question was close to determining the derivative of the function of dual variable, when we consider matrices isomorphic (algebraically and topologically) to dual numbers: $$(a+\...
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What is the matrix representation of the derivative of the vector transpose

From the definition of the Fréchet derivative and the linearity of the transpose it’s clear that the derivative of the vector transpose is the vector transpose itself. $$f(x + h) = f(x) + D(x)h + o(\...
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Jacobian of frechet derivative

Let $f:\mathbb R^n \to V$ with $V$ be a Banach space (over $\mathbb R$) suppose that $f$ is frechet differentiable in $\mathbb R^n$ then there is a way to define the Jacobian of $f$? I try to found ...
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Calculation of the Fréchet derivative

Let $E$ be normed space and $I: E \to \mathbb{R}$ be an operator given by $$I(u) = \int_{\Omega} \dfrac{|\nabla u|^{p(x)}}{p(x)} \, \mathrm{d}x,$$ where $p \in C^{0,1}(\overline{\Omega})$. Knowing ...
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Frechet differential and implicit theorem

Given: $ \Omega \subseteq \mathbb{R}^n $. $ f: \Omega \rightarrow \mathbb{R}^n $ and $ g: \Omega \rightarrow \mathcal{L}(\mathbb{R}^n, \mathbb{R}^n) $ are two $ C^1 $-class functions on $ \Omega $. ...
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Frechet derivative of "power function"

Let $H$ be a Hilbert space with norm $\|\cdot\|$, and let $p\geq2.$ Define $F(x):=\|x\|^p,\,\forall x\in H.$ Now I want to calculate the first and second order Frechet derivatives of $F$. When $p=2$, ...
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Derivative of $Ax x^\top A$ with respect to $x$

I do not want to use index notation. I want to compute the derivative $$ D_x (Axx^\top A) = ? $$ where $A$ is an $n\times n$ symmetric matrix and $x$ in a vector in $\mathbb{R}^n$. I tried resources ...
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About Symmetrie on Frechet differential.

I am trying to solve this problem. Let $E$,$F$ be Banach spaces and $f:E\to F$ be an n-times differentialble function on $a \in E$ $D^nf(a)$ with $n\geq 2$ is a multilinear, bounded and symmetrical ...
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Gateaux Diff and Frechet diff

yesterday I took a test and I can't answer this question. “Let $E,F$ be two normative spaces and f be a function on E to F with Gateaux differential and all limits $$\lim_{t \to 0}\dfrac{f(x+tv)-f(x)}{...
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Characterization of Optimal Payoff (under Expected Utility) via Gateaux-Derivative/Fréchet Derivative

Background: Let $(\Omega, \mathcal{F}, \mathbb{P})$ model a financial market and $T>0$. Denote by $(S_t)_{t\in[0,T]}$ the price process of the risky asset in the financial market. Assume that the ...
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How do I make sense of the total derivative in the limit case of $\Bbb R \to \Bbb R$ functions?

In my notes it is stated as a proposition that the total derivative of a linear map $T: V \to W$ at every point $v \in V$ is T itself: $DT(v)=T$. It also says that in the particular case of $\Bbb R \...
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What is the Frechet derivative of $\cos(x(\cdot))$?

What is the Frechet derivative of $\cos(x(\cdot))$? Let $f: C[0,1] \rightarrow \mathbb{R}$, $f(x(\cdot))$. My approach: by the definition of Frechet derivative we need to consider $$ f((x+h)(t)) - f(...
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Fréchet differentiability of a function

I am interested in determining whether the function $F:L^{\infty}(0,\infty;L^{\infty}(0,1)) \to \mathbb R$ defined by $$u \mapsto F(u)=\int_0^{\infty} \int_0^1 u(x,t)^2 \ \mathrm dx \mathrm dt $$ is ...
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Uniform Taylor expansion

$f \colon \mathbb R^n \to \mathbb R$ is differentiable in $x_0$ if there exists a functional $L$ $$f(x_0+h)-f(x_0)-Lh=o(|h|),$$ as $|h|\to 0.$ Here $o(|h|)$ denotes a function going to $0$ faster ...
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Meaning of $[f,g]$ in Frechet derivative $d^{2}C[f,g]$

Let $C$ be a quadratic functional operator of single variable. For example: $C(u)=u^{2}(t)$. The notation $dC$ means Frechet derivative. Although I know that $d^{2}C$ would be a constant since $C$ is ...
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Lagrange Multiplier for functionals

I'm reading the book called "Optimization by Vector Space Methods" by David G. Luenberger. In the proof of the Lagrange Multiplier theorem, I don't understand the last part of the statement ...
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From partial derivative of real-valued function to Fréchet derivative of Banach-space valued function

Consider a bivariate, real-valued function $u \colon X \times Y \to \mathbb{R}$, with $X, Y \subset \mathbb{R}$. Assume that $u$ is differentiable in the second variable on $X \times Y$, hence $\...
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Second order approximation of a differentiable functional

I would like to solve the following problem If $J$ is a functional twice differentiable from a normed space to $\mathbb{R}$, prove that $$ J(u+w) = J(u) + J’(u)w + \frac{1}{2}J’’(u)(w,w) + o(\lVert ...
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Gateaux derivative and stricly differentiable function

F:X->Y , suppose that F is Gateaux differentiable at point x,and also strictly differentiable at point x,then F is continiously differentiable at point x
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Difference between Gateaux and Fréchet derivative

Suppose I have the operator $$T:L_2[0,1]\to L_2[0,1],$$ $$T[x(t)] = \sin(x(t)).$$ My first question: Is it true that the Gateaux derivative of this operator equals to the ordinary derivative of $\sin(...
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Derivative of matrix inverse using directional derivative formula

I'm learning about Frechet derivatives of matrices from Bhatia's Matrix Analysis. I want to compute the derivative of the inverse function using the formula for directional derivatives. I've seen on ...
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Frechet derivative of composition [closed]

Let $f:\mathbb{R} \to \mathbb{R}$ be differentiable at each $x\in \mathbb{R}$ and let $\mathcal{H} \subset \{\mathbb{R}^n\to \mathbb{R}\}$ be a reproducing kernel Hilbert space. Is it true that $f_x:\...
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Computation of Riemannian Hessian of the Stiefel manifold

I am fairly new to differential geometry, so I'm sorry if I am sloppy in my notations. In the book Optimization Algorithms on Matrix Manifolds, the Riemannian Hessian is given by definition 5.1 as: $$...
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Does derivative assigns diffrential?

So there are 3 main definitions of derivation in 3 different contexts. Calculus of one variable real functions. Say we have an everywhere differentiable function $f: \mathbb{R} \to \mathbb{R}$. Then ...
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Do 'Anti-Fréchet Derivatives' work similar to typical anti-derivatives? Are there two ways different ways to define them?

Assume a function $f:L_2(R^{+}):R$ is frechet differnetiable in $x\in L_2(R^{+})$ in that there exists a unique function $D(x_i,x)$ (where $x_i\in R^{++}$ is an element of x) such that: $f(x+h)=f(x)+\...
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Frechet derivative and Lagrangian

Consider the Lagrangian $L:\mathbb{R}\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}:x,y,z\rightarrow L(x,y,z)$ and cost functional defined by the integral of a lagrangian $$J(y) := \int_{a}^bL(...
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Fréchet differentiation of integral operator on $L^2(0,1)$ or $C[0,1]$

For an exercise (3.2.27 in P. Drábek, J. Milota, "Methods of Nonlinear Analysis"), I'm trying to differentiate (in the Fréchet sense) this operator $$ F(\varphi) = \int_0^1 \left[ \int_0^t |\...
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Differentiation and Chain Rule on the Hilbert Space $L^2$. (Reisz Representation).

Let $F:L^2(\mathbb{R}^d)\to \mathbb{R}$ be a functional on the Hilbert space $L^2(\mathbb{R}^d)$ and $\rho:\mathbb{R}\to L^2(\mathbb{R}^d)$ a curve in the space $L^2(\mathbb{R}^d)$. I want to ...
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Riemann integral in a Banach space

Let $f:[0,\infty) \rightarrow E$ be a continuous function that takes value in a Banach space $E$. I know that we can define the integral $\int^x_0 f(t)dt, \forall x \in [0,\infty)$ using Riemann ...
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Notions of derivative and integral in Banach spaces in Brezis' book

In chapter 7, page 184 of the book Functional analysis by Haim Brezis, the author gave a theorem of Cauchy, Lipchizt and Picard as follows: Theorem 7.3 (Cauchy, Lipchitz, Picard). Let $E$ be a Banach ...
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An application of chain rule to compute $\frac{\partial^2}{\partial x_k^2} g(y-x)$

Let $g: \mathbb R^d \to \mathbb R$ be twice differentiable. Fix $y \in \mathbb R^d$ and $k \in \{1, 2, \ldots, d\}$. I would like to compute $\frac{\partial^2}{\partial x_k^2} g(y-x)$. I'm sorry for ...
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Does the Frechet derivative on a separable Hilbert space commute with the projection onto subspaces?

Let $H$ be a separable Hilbert space with a fixed orthonormal basis $\{ e_n \}$ and $f : H \to H$ be a sufficiently "nice" mapping so that it has the Frechet derivative $Df \in L(H,H)$. Here,...
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How to find the Fréchet derivative of a matrix exponential?

Let $f : \Bbb R^{n\times n} \to \Bbb R^{n \times n}$ be the matrix exponential $$ f(A) = \sum_{k = 0}^\infty \frac{A^k}{k!} $$ Let $B$ be a matrix that commutes with $A$. Show that the value of ...
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Whether the following subset in the space of function is closed?

I am facing a problem as the following: Suppose $\mathcal{H}$ is a Hilbert space with inner product $\langle\cdot,\cdot\rangle$ and its induced norm $|\cdot|$. Consider the space of functions $(x,t)\...
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Chain rule for a functional derivative.

Given $L:\mathbb{R}\times \mathcal{P}(\mathbb{R}^n) \to \mathbb{R}$, where $\mathcal{P}(\mathbb{R}^n)$ is the space of probability densities on $\mathbb{R}^n$. I want to calculate $$ \frac{d}{d\...
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Derivative of a continuous bilinear form

I'm trying to solve below exercise Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space. Let $a: H \times H \rightarrow \mathbb{R}$ be a continuous bilinear form. Determine the derivative ...
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Brezis' exercise 5.14

I'm trying to solve below exercise in Brezis' Functional Analysis Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space and $\Vert \cdot\Vert $ its induced norm. Let $a: H \times H \...
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Please explain the steps involved after $F(x+h)-F(x)-BAh=Bo_1(h)+o_2(\phi(h))$ in detail

Please explain the steps involved after $F(x+h)-F(x)-BAh=Bo_1(h)+o_2\phi(h)$ in detail. Note 1:here differentiable means Frechet differentiable Source:Analysis for Applied Mathematics by WARD CHENEY(...
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Reference Needed - Taylor's Theorem with Fréchet Derivatives

According to Wikipedia, Taylor's Theorem holds for Fréchet derivatives, but no reference is given. I started looking in various books and they all mention that it is possible to write down a version ...
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Fréchet derivative of a matrix expression [closed]

Suppose $h(Q) = Q^{T} A Q$, then the Fréchet derivative is given by $D_{h} (Q) [H] = H^{T} A Q + Q^{T} A H$. I am bit unsure about this so-called Fréchet derivative is obtained. I would have just said:...
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Meaning of delta-u term in variational derivative

I am considering a question in the calculus of variations. I can understand the concept of the variational derivative, but I am not sure what the $\delta u$ in this question means: "6. Let $Ω⊆R^n$...
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Fréchet derivative of inverse function is automatically continuous

Suppose $X$ and $Y$ are Banach spaces and $U\subseteq X$, $V\subseteq Y$ are open subsets. Let $f:U\to V$ be bijective and continuously Fréchet differentiable with derivative $Df:U\to\mathcal{L}(X;Y)$....
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Is $n$-times differentiable equivalent to $n$-times Fréchet-differentiable for functions from $\mathbb C$ or $\mathbb R$ to a Banach space?

In the following, $\mathbb K$ denotes $\mathbb C$ or $\mathbb R$ and $E$ is a $\mathbb K$-Banach space I have known that for a function $f:\mathbb K\supseteq X\to E$ is called differentiable at $a$ ...
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