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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Computing Frechet derivative
Let $F:C^{([-1,1])}\rightarrow C^{([-1,1])}$ defined as $F(q)(x)=q^{2}(-x)$.
I want to find the derivative of this function my current stratergy is to use the directional derivative so $dF(q)(x)h= \...
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Differentiability at the origin of Norm
For $\alpha > 1$, show that $f:\mathbb{R}^n \to \mathbb{R}^n$, given by
$$f(x)=\rVert x\rVert^\alpha$$
is differentiable at the origin.
What I did:
What is necessary is to find a linear ...
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1answer
30 views
Understanding Differentiability of an Inner Product map
I have been trying to solve the following question without success.
Question:
Show that $f:\mathbb{R^n} \times \mathbb{R^n}\to \mathbb{R}$, defined as
$$f(x,y) = x\cdot y$$ is differentiable and ...
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Is the functional $w \mapsto \int_0^1 | \ \| w(t) \| - 1 | \ dt$ $C^1$ or even smooth?
Let $H:= H(I;\mathbb{R}^3)$ be the space of $L^2$ + absolutely continuous functions with $L^2$ derivative.
For $w \in H$ consider the functional
$$\psi(w) = \int_0^1 | \ \| w(t) \|^2 - 1 | \ dt$$
...
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L'Hospital Rule for Vector Valued Functions
Question:
Using Rudin's formulation of the L'Hospital Rule, show that it remains true if
a) $f:(a,b)\to \mathbb{R^k}$ is vector-valued and $g:(a,b)\to \mathbb{R}$
b) $f:(a,b)\to \mathbb{C}$ and $g:(...
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How to prove that Frechet derivative exists and coincides with the Gateau derivative?
Let $X$ and $Y$ be Banach spaces and $U \subseteq X$ be open.Let $F:U \to Y$ be Gateaux differentiable and let the mapping $x \to F′(x)$ be continuous from $U \in L(X,Y)$. How can I prove that the ...
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Fréchet Derivative of functionals which depend on the derivative
Let $\Omega:=[x_a,x_b]\times[t_a,t_b]\subset \mathbb{R}^2$ and consider the functional $D:C^2\left(\Omega,\mathbb{R}\right)\rightarrow \mathbb{R}$ such that
$$D[y]= \iint_{\Omega}d(x,t,y) \, dx \, dt$$...
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Existence of continuous derivatives of partial functions and total differentiability
We know that for a function $\mathbb{R}^m\to \mathbb{R}^n$ the existence and continuity of partial derivatives implies the differentiability of that function.
Will this hold true for functions on ...
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Partial Derivative of Frechet-differentiable Function
Let $X,Y,Z$ be Banach-Spaces and $F:X \times Y \rightarrow Z$ Frechet-differentiable. Then it holds
$$ F'(x,y)(u,v) = F_x (x,y)u+ F_y(x,y)v .$$
How do I prove this?
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What does the reparameterization mean in Fréchet distances?
I am trying to understand the definition of frechet distance but I am struggling to understand the reparameterization part in the definition. I got the following definition from wikipedia
Let A and ...
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1answer
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Help showing derivative is compact
I'm struggling with proving the following theorem:
$\textit{Let }A:U\subset X\rightarrow Y\textit{ be a completely continuous operator from an open subset U of a normed}$
$\textit{space X into a ...
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Calculation of Gateaux Derivative
I would like to ask some details for the calculation (and the justification) of the following problems.
Let us denote $u^{+} =\max\{u,0\}$. Take $u,v \in H_{0}^{1}(\Omega)$ for a bounded domain $\...
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Why is there an “implication” rather than and “and” in this definition of the derivative?
I am readig Pugh's Analysis book:
Definition
Let $f:U \to \mathbb{R}^m$ be given where $U$ is an open subset of $\mathbb{R}^n$. The function $f$ is differentiable a $p \in U$ with derivative $(...
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Frechet derivative of a function on matrix-valued $L^2$ functions
$\newcommand{\tr}{\text{Tr}}$
Let us denote $L^2(X, \mathbb C^{n \times n})$ be the space of matrix-valued $L^2$ functions. That is, if $f \in L^2(X, \mathbb C^{n \times n})$, each entry of $f$ is an $...
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How to find Frechet derivative of $f(x)=\|Ax-b\|^2$ at any $x^*$?
Given a real $m \times n$ matrix $A$ and $b \in \mathbb{R}^m$, let $f(x)=\|Ax-b\|^2$ for any $x \in \mathbb{R}^n$.
Find Frechet derivative of $f(x)=\|Ax-b\|^2$ at any $x^*$?
Actually I am wondering ...
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Showing that a given function is Frechet differentiable
I'd like to do the following exercise from my book. The statement is as follows.
Let $\gamma \epsilon { C }^{ 1 }(R\times { R }^{ n })$,$c>0$ and ${ \gamma }_{ 0 }\epsilon { L }^{ 1 }({ R }^{ n })...
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2answers
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Showing a function is Frechet Differentiable？
I just started learning the Frechet Derivatives.
So I have a function $H:\mathbb{R}^{N\times n}\to\mathbb{R}^{N\times n}$, i.e. $U^T\in\mathbb{R}^{N\times n}$ and
$$H(U^T)=GW\times (F(U))^T+S\times U^...
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Derivative of a functional with respect to another functional
I am trying to make sense of functional derivatives and have a couple of questions bothering me:
Let $F[X]$ be a functional of $X(t)$ and $G[X]$ another functional of $X(t)$.
By chain rule in the ...
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Fréchet derivative: dependency of the choice of norm
Let $ f_1: L^2(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{L^2 (\Omega)}^2 $, $ f_2: H^1(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{L^2 (\Omega)}^2 $ and $f_2: H^1(\Omega)\...
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Does $[F'(u^*)]^{-1}$ exist if $u^*$ is a solution of $\partial_t u + [u+1-t] \partial _x u -1 = 0$?
Given
$$F(u):=\partial_t u(x,t) + [u(x,t)+1-t] \partial_x u(x,t) -1 $$
we can calculate the derivative of $F$ and get that the derivative of $F'(u)$ applied to $v$ gives us
$$F'(u)(v)= \partial_t ...
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minimize norm $||C-A^TB ||^2$ by taking derivative with respect to vector
Let's try to minimimize a matrix function $f(B)=||C-A^TB ||$ with respect to $B$. Here $C_{k \times 1}$ and $A_{3 \times k}$ and $B_{3 \times 1}$ matrices. One solution presented to me was to take ...
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1answer
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Computing Fréchet derivative of $F(f)(x) = \int^{x}_{0} \cos(f(t)^{2})dt, x \in [0,1]$
Let $X = \mathcal{C} \left( [0,1] \right)$ be the Banach space of continuous functions on $[0,1]$ (with the supremum norm) and define a map $F : X \rightarrow X$ by
$$F(f)(x) = \int^{x}_{0} \cos(f(...
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1answer
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Proving the product rule for Fréchet Derivative.
Let $X$ be a normed vector space, $U\subset X$, and $F,G:U\rightarrow \mathbb{R}$ differentiable at $x\in U$. Show that the map $F\cdot G:U\rightarrow \mathbb{R}$, $F\cdot G(x)=F(x)G(x)$ is also ...
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Frechet derivative of basic derivative
I was reading about functional analysis and it interested me greatly applying the techniques of analysis to function spaces but it seems to have abruptly stopped when getting to the derivative. I know ...
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Higher total derivatives (Frechét derivatives)
This issue I just cannnot resolve, so I'd highly appreciate your help.
Let $a_1, ... , a_n \in \mathbb{R}^k$ with $k$ a natural positive number. If we consider the function $$ W: \mathbb{R}^k \to \...
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1answer
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Fréchet derivative; open set
I have some problems in understanding the definition of the Fréchet derivative of an operator $F: X \rightarrow X$. In fact, most authors report that $F$ must be defined in some open neighborhood of ...
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1answer
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Directional derivatives of matrix trace functionals
Let $P$ be an $n\times n $ positive semidefinite matrix over $\mathbb{C}$, let $p\in\mathbb{R}$ be in the range $0<p<1$. Consider the function $g:[0,\infty)\rightarrow\mathbb{R}$ defined by $g(x)...
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1answer
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Explaination of the Frechet Derivative
Let $E$ and $F$ be Banach spaces and let $U$ be an open subset of $E$. Suppose $g:U \to F $. $g$ is continuous at $x_0$ if there exists a linear transformation, $T_{x_0}$, such that $$ \lim \limits_{...
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The matrix $I_n - v^t x$ is invertible when $\langle v, x \rangle \neq 1$
Let $v \neq 0$ be any vector in $\Bbb R^n$ and let, $U_v = \{x \in \Bbb R^n : \langle v,x \rangle \neq 1\}$. Then:
(i) Show that the matrix $I_n - v^t x$ is invertible $\forall x \in U_v$.
(...
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Lagrange multiplier method on Banach spaces (in Dirac notation)
I want to prove Cauchy–Schwarz' inequality, in Dirac notation, $\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2$, using ...
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Application of Higher Order Derivatives
If $V = \mathbb{R}^{n}$ and $W = \mathbb{R}^{m}$, and $f:V\rightarrow W$ is smooth, then higher order derivatives at $x\in V$, $D^{n}_{x}f$ can be thought of as symmetric, multilinear functions $V\...
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The relation between Frechet derivative and differentiation in $\mathbb{R}^2$
In the book of Complex Made Simple by Ullrich, at page 4, it is given that
However, if we are considering $\mathbb{C} = \mathbb{R}^2$, then $Df$ is a $2\times 2$ matrix, and not a complex number, but ...
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1answer
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If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b $ [closed]
Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then ...
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1answer
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Linearity of the derivative operator of polynomials
Suppose $f(x)=x^i$ as a simple polynomial function of power $i$. As we know from the definition of the derivative operators, such an operator must be linear. How can we prove that the derivative ...
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“FTC”-type relationship between Frechet Derivative and Bochner integral?
Let $X,Y$ be Banach spaces and $(X,\Sigma,\mu)$ be a measure space. Then, for a function $f: X\to Y$ under suitable conditions, we can define the Frechet derivative $\text df: X\to B(X,Y)$ or $\text ...
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An exercise on the calculation of some Fréchet derivatives.
The following is Exercise VII.5.15.(iii) from Analysis II by Amann & Escher.
Suppose $X$ is open in $\mathbb{R}^m$. For $f,g\in C^1(X,\mathbb{R}^m)$, define $[f,g]\in C(X,\mathbb{R}^m)$ by $$[f,...
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Notion of derivative used in Petersen & Pedersen's Matrix Cookbook
I am looking at the Matrix Cookbook. From my real analysis background, my understanding of calculating derivatives involving matrices is to use the Fréchet derivative on the normed space $(\mathbb{R}^{...
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1answer
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Chain rule for higher Fréchet derivatives?
I'm having trouble with the proof of the following fact.
Let $E,F,G$ be Banach spaces. Suppose $X$ is open in $E$ and $Y$ is open in $F$. Given functions $f\in C^m(X,F)$, $g\in C^m(Y,G)$ such that $...
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2answers
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Problem with understanding higher Fréchet derivatives and multilinear maps.
Let $E,F$ be Banach spaces, $X$ an open set in $E$, $x_0\in X$ and $f:X\to F$. Then the $n$th Fréchet derivative $\partial^nf(x_0)$ of $f$ at $x_0$ is an $n$-linear map in $\mathcal{L}^n(E,F)$. So ...
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1answer
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Is matrix exponentiation $\exp:\mathcal{L}(E)\to\mathcal{L}(E)$ continuously Fréchet differentiable?
This is exercise VII.3.8 in Amann & Escher, Analysis II.
Let $E$ be a Banach space. Denote by $\mathcal{L}(E)$ the space of all bounded linear maps from $E$ to itself. Then the exponential map $...
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Is the Fréchet derivative of a measurable function measurable?
Let
$(\Omega,\mathcal A)$ be a measurable space
$E,F$ be $\mathbb R$-Banach spaces
$f:\Omega\times E\to F$ such that $f(\;\cdot\;,x)$ is strongly $(\mathcal A,\mathcal B(E))$-measurable for all $x\in ...
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Is there a good reference for Hölder spaces of Fréchet differentiable functions between Banach spaces?
The notion of Hölder continuity can be defined for any function between metric spaces. However, the topological properties of the space generated by continuously differentiable functions whose ...
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1answer
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A polynomial map between Banach spaces is Fréchet differentiable
I am reading up on the proof of the following statement:
Let $p:\mathbb{X} \rightarrow \mathbb{Y}$ be a polynomial. Then $p$ is Frechet differentiable at every $x \in \mathbb{X}$, with $$dp(x) v = ...
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1answer
103 views
Lipschitz function which is Gâteaux-differentiable is Fréchet-differentiable
Let $U \subset \mathbb{R}^n$ open set, $x_0 \in U$ and $f : U \rightarrow \mathbb{R}$ a function. Show that if $f$ is Lipschitz and Gâteaux-differentiable at point $x_0$, then $f$ is Fréchet-...
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Frechet differentiability of inverse mapping
Let $f: V \rightarrow W$ be a homeomorphism (where $V$, resp. $W$, is an open set of a Banach space $E$, resp. $F$); assume $f$ is differentiable at a point $a \in V$. In order that $g = f^{-1}$ be ...
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4answers
263 views
Frechet derivative of a composition of functions over matrices
In control theory, the discrete Lyapunov equation is defined as
\begin{align*}
A^T X A + Q = X,
\end{align*}
where $A \in \mathcal{M}(n \times n; \mathbb R)$ and $Q \in \mathbb {S}_{++}$ ( positive ...
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0answers
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Finding a functional satisfying a given Frechet derivative
Usually, we are interested in finding the Frechet derivative of a given functional. My problem is the opposite; to find a functional satisfying a condition given in terms of the Frechet derivative of ...
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1answer
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Proof about the derivative of operators in $L(V,V')$
I'm trying to understand the following proposition (from the textbook "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations"-Showalter, chapter III.3, proposition 3.1):
...
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What's the second Fréchet derivative of the tensor product of two Fréchet differentiable functions?
Let
$X,Y,E$ be $\mathbb R$-Banach spaces
$f:E\to X$ and $g:E\to Y$ be twice Fréchet differentiable
What's the second Fréchet derivative of $f\otimes g$?
$f\otimes g$ has to be understood as the ...
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0answers
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What is the second Fréchet derivative of $x\mapsto x\otimes x$?
Let
$T>0$
$I:=(0,T]$
$\mathcal B(I)$ denote the Borel $\sigma$-algebra on $I$
$E$ be a $\mathbb R$-Banach space
$\mu:\mathcal B(I)\to E$ be a vector measure
$X:I\to E$ be bounded and $(\mathcal B(...
