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.

Filter by
Sorted by
Tagged with
1 vote
1 answer
38 views

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 ...
Est's user avatar
  • 11
1 vote
1 answer
37 views

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)}{...
Esteban LS's user avatar
2 votes
0 answers
21 views

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 ...
MWilk's user avatar
  • 53
0 votes
4 answers
102 views

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 \...
some_math_guy's user avatar
0 votes
0 answers
48 views

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(...
wxist's user avatar
  • 471
0 votes
0 answers
36 views

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 ...
elmas's user avatar
  • 111
4 votes
1 answer
101 views

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 ...
carlos85's user avatar
  • 123
0 votes
0 answers
13 views

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 ...
Redsbefall's user avatar
  • 4,835
0 votes
1 answer
42 views

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 ...
Lee's user avatar
  • 143
2 votes
0 answers
39 views

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 $\...
Guran Semiotovic's user avatar
2 votes
1 answer
52 views

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 ...
coboy's user avatar
  • 1,195
0 votes
0 answers
20 views

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
VadimStacheff's user avatar
1 vote
0 answers
81 views

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(...
VadimStacheff's user avatar
0 votes
1 answer
67 views

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 ...
Afham's user avatar
  • 27
0 votes
1 answer
70 views

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:\...
wzell's user avatar
  • 67
2 votes
1 answer
120 views

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: $$...
Cristian Emiliano Godinez's user avatar
4 votes
0 answers
80 views

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 ...
Andrey's user avatar
  • 103
2 votes
0 answers
21 views

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)+\...
Vance M's user avatar
  • 112
1 vote
0 answers
68 views

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(...
R. C.'s user avatar
  • 21
0 votes
0 answers
54 views

Differentiability of the limits of sequences of functions between Banach spaces

$\newcommand{\vertiii}[1]{{\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert #1 \right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert}}$Let $(E, |\cdot|_E)$ be a real Banach space and $X$ an ...
Akira's user avatar
  • 17.2k
2 votes
1 answer
115 views

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 |\...
david_sap's user avatar
  • 444
2 votes
2 answers
93 views

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 ...
jay's user avatar
  • 187
2 votes
1 answer
76 views

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 ...
Jeffrey Jao's user avatar
2 votes
1 answer
118 views

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 ...
Jeffrey Jao's user avatar
0 votes
0 answers
35 views

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 ...
Analyst's user avatar
  • 5,617
0 votes
0 answers
55 views

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,...
Keith's user avatar
  • 7,411
0 votes
0 answers
112 views

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 ...
Jimmy's user avatar
  • 1
3 votes
0 answers
49 views

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)\...
LonerSeekDefeat's user avatar
1 vote
1 answer
104 views

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\...
hoh hoh's user avatar
  • 11
0 votes
0 answers
54 views

Continuity of evaluation of Frechet derivative on function of finite signed measures

Disclaimer: This is not a homework problem, so full answers/explanations are appreciated. Setup: Let $S$ be a second-countable compact Hausdorff space, and let $B(S)$ denote its Borel $\sigma$-algebra....
brenderson's user avatar
  • 1,452
2 votes
0 answers
52 views

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 ...
Akira's user avatar
  • 17.2k
2 votes
0 answers
55 views

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 \...
Akira's user avatar
  • 17.2k
0 votes
0 answers
28 views

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(...
Anwar's user avatar
  • 119
1 vote
1 answer
280 views

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 ...
Euler_Salter's user avatar
  • 5,061
0 votes
1 answer
167 views

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:...
user21369645's user avatar
1 vote
0 answers
28 views

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 $Ω⊆R^n$...
therickster's user avatar
0 votes
0 answers
23 views

Show that $f$ belongs to $C^m(X, F)$ if and only if $f$ is $m$-times continuously partially differentiable

Suppose $X$ is open in $\mathbb R^n$, $f:X\to F$, and $m\in\mathbb N^+$. Here the derivative is a bounded linear map, it is endowed with operator norm. I can prove the situation of $m=1$ and $2$, but ...
HsiL's user avatar
  • 11
1 vote
0 answers
81 views

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 $Df:U\to\mathcal{L}(X;Y)$....
junjios's user avatar
  • 1,352
0 votes
0 answers
51 views

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 $a$ ...
HsiL's user avatar
  • 11
1 vote
0 answers
56 views

Shouldn't the definition of the Frechet derivative not require $a$ to be a limit point?

Throughout this section we assume $V$ and $W$ are normed vector spaces, with $V'$ an open subset of $V$ and $f$ a function defined $V'\to W.$ Definition: The function $f:V'\to W$ is differentiable at $...
Sam's user avatar
  • 4,680
1 vote
0 answers
85 views

Functional derivative of $u \mapsto R\left(\int_{\Theta} \phi(\theta) \; \text{d}u(\theta)\right) + \lambda u(\Theta)$ for matrix-valued measure $u$

Let $\Theta$ be a closed connected manifold, $H$ and $F$ Hilbert spaces and $R \colon H \to \mathbb R$ and $\phi \colon \Theta \to F$ smooth. What is the functional derivative of $$f(u) := R\left(\...
ViktorStein's user avatar
  • 4,788
2 votes
1 answer
85 views

Does existence and continuity of partial derivatives imply differentiability in Normed Vector Spaces?

I'm wondering if the 'Normed Vector Space version' of the following theorem holds: Theorem: let $A\subseteq \mathbb{R}^n$ be open and let $f:A\to \mathbb{R}^m$ have continuous partial derivatives $\...
Sam's user avatar
  • 4,680
4 votes
1 answer
66 views

The integral of a small-o function is small-o

In the question on Frechet derivative it is used that if $g:\mathbb{R} \to \mathbb{R}$ is a continuous function s.t. $g(x) \in o(x)$, i.e. if $\lim_{x \to 0} \frac{|g(x)|}{x}=0$ then $$ \int_0^1g(h(t))...
Abm's user avatar
  • 291
1 vote
1 answer
64 views

The meaning of the notation $\frac{\mathrm d \gamma^\chi}{\mathrm d t}(0)$

I'm reading about how a smooth curve is related to a tangent vector from this thread, i.e., Given an $m$-dimensional manifold $M$ and $p \in M$, a tangent vector of $M$ at $p$ is any function $$ v:\{\...
Analyst's user avatar
  • 5,617
2 votes
1 answer
131 views

An example of $f$ and $a$ such that $f$ is Gâteaux but not Fréchet differentiable at $a$

I'm reading this lecture note about differentiability. Let $(X, |\cdot|_X)$ and $(Y, |\cdot|_Y)$ be normed spaces. Let $A$ be an open subset of $X$ and $f: A \to Y$. The directional derivative $f^{\...
Akira's user avatar
  • 17.2k
1 vote
0 answers
47 views

What is the relationship between almost everywhere differentiable and Frechet differentiable???

Suppoce for every $u$, $f(u)$ is a square integrable function, namely, $f(u)\in L^2([0,1])$. We can compute the Frechet derivative of this functional $f$, or in other words, the condition for $f$ to ...
八百标兵's user avatar
1 vote
1 answer
150 views

Is a function composition Fréchet differentiable?

I have a trouble in handling a following question. Given a composition operator $ F :C^1([0,1]) \to C^1([0,1]) ; F(f) = f(f(x))$, I have a Gâteaux derivative of $F$. \begin{align} \lim_{t \to 0} \...
yeachan park's user avatar
1 vote
1 answer
118 views

Show that the Lagrangian $\int_I H(t,c(t),c^\prime(t)) dt$ is Frechet differentiable

Consider an interval $I = [t_0,t_1]$ and a finite dimensional Banach space $X$. Let $U$ be an open subset of $\mathbb{R} \times X \times X$ and let $V \subseteq \mathcal{C}^{1}(I,X)$ be the set of ...
3nondatur's user avatar
  • 4,084
0 votes
1 answer
57 views

Frechet derivative of a special function

Let $\Phi (u)=\tilde{M}\Big( \int_{\Omega}\frac{1}{p(x)}(|\nabla u(x)|^{p(x)}+\alpha (x)|u(x)|^{p(x)})dx\Big)$ where $u\in W^{1,p(x)}(\Omega)$ (the generalized Sobolev space), $\Omega \subset \mathbb{...
M.Ramana's user avatar
  • 2,743
0 votes
0 answers
48 views

How to find the Frechet derivative of $f: V \rightarrow \mathbb{R}, c \mapsto \int H(t,c(t),c^\prime(t)) dt $?

Consider an interval $I = [t_0,t_1]$ and a finite dimensional Banach space $X$. Let $U$ be an open subset of $\mathbb{R} \times X \times X$ and let $V \subseteq \mathcal{C}^{1}(I,X)$ be the set of ...
3nondatur's user avatar
  • 4,084

1
2 3 4 5
10