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 ...
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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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1answer
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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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37 views
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
208 views
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
$$H(U^T)=GW\times (F(U))^T+S\times U^...
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35 views
Derivative of a functional with respect to another functional
I am trying to make sense of functional derivatives and have a couple of questions bothering me:
Let $F[X]$ be a functional of $X(t)$ and $G[X]$ another functional of $X(t)$.
By chain rule in the ...
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28 views
Fréchet derivative: dependency of the choice of norm
Let $ f_1: L^2(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{L^2 (\Omega)}^2 $, $ f_2: H^1(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{L^2 (\Omega)}^2 $ and $f_2: H^1(\Omega)\...
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Fréchet derivative of $W \in C^2(\Omega \times Q)$
I'm a reading a book about PDE. There is a function $W: \Omega \times Q \to \mathbb{R}$, where $\Omega \subseteq \mathbb{R}^3$ is an open set and $Q$ is the set of $3\times 3$ matrices with positive ...
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Does $[F'(u^*)]^{-1}$ exist if $u^*$ is a solution of $\partial_t u + [u+1-t] \partial _x u -1 = 0$?
Given
$$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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38 views
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
54 views
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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35 views
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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1answer
70 views
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 $$ W: \mathbb{R}^k \to \...
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1answer
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Fréchet derivative; open set
I have some problems in understanding the definition of the Fréchet derivative of an operator $F: X \rightarrow X$. In fact, most authors report that $F$ must be defined in some open neighborhood of ...
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1answer
138 views
Directional derivatives of matrix trace functionals
Let $P$ be an $n\times n $ positive semidefinite matrix over $\mathbb{C}$, let $p\in\mathbb{R}$ be in the range $0<p<1$. Consider the function $g:[0,\infty)\rightarrow\mathbb{R}$ defined by $g(x)...
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37 views
Explaination of the Frechet Derivative
Let $E$ and $F$ be Banach spaces and let $U$ be an open subset of $E$. Suppose $g:U \to F $. $g$ is continuous at $x_0$ if there exists a linear transformation, $T_{x_0}$, such that $$ \lim \limits_{...
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3answers
67 views
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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79 views
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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39 views
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
72 views
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
51 views
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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26 views
“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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1answer
111 views
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
73 views
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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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
55 views
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
82 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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22 views
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
250 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 ...
3
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0answers
67 views
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
43 views
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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38 views
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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67 views
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(...
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25 views
Are derivatives of functions which depend only on direction determined by their values on the sphere?
(Some text in this question is duplicated from another related question of mine.)
Let $f:\mathbb{R}^p \to \mathbb{R}$ be smooth function and define the function $g: x \mapsto f[x(x^Tx)^{-1/2}].$ ...
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0answers
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Frechet derivative of a characteristic fucntion
Let $\delta$ be a small positive number, and $x\in [0,1]$, $r\in D(B):= [\delta,1-\delta]$. Consider the function:
$$b(x;r)=\begin{cases}1 \quad |x-r|<\delta\\ 0 \quad elsewhere \end{cases}$$
For ...
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0answers
55 views
Are there simple conditions on a Fréchet differentiable function on a Hilbert space that ensure that the Fréchet derivatives are trace-class?
Let $H$ be a separable $\mathbb R$-Hilbert space and $f:H\to H$ be Fréchet differentiable. Are there "simple" conditions on $f$ that ensure that the Fréchet derivatives ${\rm D}f(x)\in\mathfrak L(H)$ ...
2
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1answer
55 views
Prove that $f$ is linear
Let $f: \mathbb{R}^{m} \longrightarrow \mathbb{R}$ differentiable such that $\displaystyle f\left(\frac{x}{2}\right) = \frac{f(x)}{2}$ for all $x \in \mathbb{R}^{m}$. Prove that $f$ is linear.
I'm ...
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1answer
62 views
Calculating a Frechet derivative of a function of functions
I have been attempting this problem for quite a while. The problem is to show that the following function $$F((f))(x)=\int_{0}^{x}f^2(t)\cos(t)dt$$ is differentiable on the normed vector space $C[0,1]$...
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1answer
18 views
Frechet derivative of an homogeneous function
Let be $ f:X\to X$ a homogeneous function of degree 1 definied on a real Banach space $X$ and suppose that exists $u\in X$ such that $f(u)=u$. Let be $L$ the Frechet derivative of $f$ at $u$, i.e. $L=...
3
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2answers
51 views
How to show that $\Psi: E \rightarrow E$, $\Psi(f) = \sin(f(t))$ is continuous and differentiable?
Let $E = \mathcal{C}([0,1],\mathbb{R})$ a Banach space of continuous fonctions mapping from $[0,1]$ to \mathbb{R}, with the norm $||f|| = \underset{t \in [0,1]}{\sup}|f(t)|$. Let $\Psi : E \rightarrow ...
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1answer
69 views
Prove $\lim_{h \to 0^{+}}\frac{\lVert u +hv \rVert_{\infty} - \lVert u \rVert_{\infty}}{h}=\max_{x \in M}(v\cdot \operatorname{sign}(u))$
I have posted this question before but did not get any answer so im trying again:
Let $u,v \in C[a,b]$ and $M=\{x \in [a,b]: \lVert u \rVert_{\infty} = |u(x)| \}$. The Gâteaux derivative of the ...
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0answers
25 views
If $ \|G(x+ty)\|<\|G(x)\| $, is then $ \|G(x) + tG'(x)[y]\| <\|G(x)\| $?
It seems intuitive that given a $ C^1 $ function $ f : \mathbb R \to \mathbb R $, where
for some $ t_0 > 0 $ we have $ |f(x+t)| < |f(x)| $ for all $ t \in (0,t_0) $, and
$ f'(x) \neq 0 $,
...
0
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1answer
66 views
Fréchet derivative of matrix-valued function
If matrix-valued function $f : \mathbb{X} \times \mathbb{X} \to \mathbb{X}$ is defined by $(X, Y) \mapsto XY$, then how can we calculate the Fréchet derivative using the definition?
Note that $\...
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1answer
120 views
Notion of continuous partial derivatives in Banach spaces
Really simple question: If $X,Y,Z$ are Banach spaces, what's meant if we say that a function $f:X\times Y\to Z$ is continuously Fréchet differentiable in the first variable?
The Fréchet derivative in ...
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0answers
97 views
Proof verification + help on last step - Fréchet Differentiable of bilinear function
Problem: Let $X$ and $Y$ be two normed linear spaces. A function $ϕ : X×Y → \mathbb{R}$ is bilinear
if $f(·, y) ∈ L(X, \mathbb{R})$ and $f(x, ·) ∈ L(Y, \mathbb{R})$ for each $(x, y) ∈ X × Y$ . Show ...
1
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1answer
62 views
Proof of Fréchet Differentiability - general instruction and specific problem
I'd like to know if anyone could provide a general instruction to proving that a function is Fréchet differentiable. I've run into some problems where usually I can propose a form for the linear ...
