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 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
$$f(u) := R\left(\...
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Does existence and continuity of partial derivatives imply differentiability in Normed Vector Spaces?
I'm wondering if the 'Normed Vector Space version' of the following theorem holds:
Theorem: let $A\subseteq \mathbb{R}^n$ be open and let $f:A\to \mathbb{R}^m$ have continuous partial derivatives $\...
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The integral of a small-o function is small-o
In the question on Frechet derivative it is used that if $g:\mathbb{R} \to \mathbb{R}$ is a continuous function s.t. $g(x) \in o(x)$, i.e. if $\lim_{x \to 0} \frac{|g(x)|}{x}=0$ then
$$
\int_0^1g(h(t))...
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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:\{\...
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In which cases is a Gateaux differentiable function also a Fréchet differentiable function? [closed]
It is well known that a function which is Fréchet differentiable is also Gateaux, but in which conditions can we be sure that the opposite is also true? And in that case, is there any proof of that?
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Proof of The chain rule for Normed Vector Spaces
So I'm trying to prove the chain rule over $2$ normed vector spaces, namely
$f: \Omega \subseteq E \to V \subseteq F$ and $g:V\subseteq F \to G$ differentiable at $a\in \Omega$ and $f(a) \in V$ ...
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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^{\...
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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 ...
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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} \...
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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 ...
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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{...
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Gâteaux derivative of a special function
Let $X$ and $Y$ be normed linear spaces and let $f:X\to Y$. If for $a,h\in X$ the limit (in the norm of $Y$) $$\lim_{t\to 0}\frac{f(a+th)-f(a)}{t}$$ exists, then its value is called the derivative of $...
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How to find the Frechet derivative of $f: V \rightarrow \mathbb{R}, c \mapsto \int H(t,c(t),c^\prime(t)) dt $?
Consider an interval $I = [t_0,t_1]$ and a finite dimensional Banach space $X$. Let $U$ be an open subset of $\mathbb{R} \times X \times X$ and let $V \subseteq \mathcal{C}^{1}(I,X)$ be the set of ...
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Explanation of 'little o' notation in the context of Fréchet derivatives
Consider the function $x^{2}$. Its Fréchet derivative should be $2x$ which is confirmed by evaluating the definition of Fréchet derivative
$$
\lim_{|h|\rightarrow0}{\frac{(x + h)^2 - x^{2}- 2xh}{|h|}=\...
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Fréchet derivative with composition of norm
I'm trying to understand an argument to compute a Fréchet derivative. Someone could help me to figure out? I don't see where it comes from the 2nd equality. The first one it's only use the identity ...
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Chain rule in a Hilbert space.
Let $F:H\to \mathbb{R}$ be some functional on a Hilbert space $H$. Denote its Frechet derivative at $h\in H$ as $\frac{\delta F}{\delta h}(h)$. Suppose $h_t$ is a curve in $H$ i.e $$h_\cdot : \mathbb{...
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Lagrange multiplier in Banach space
I am trying to understand a proof of the Lagrange multiplier theorem for Banach spaces and there is some point I do not understand. Let me recall the setting.
Let $I, J: E \to \mathbb R$ two ...
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Is the sup norm a differentiable function on $\mathcal{C}[a,b]$?
Is the sup norm a differentiable function on $\mathcal{C}[a,b]$?
Hint: Consider $F:[a,b] \rightarrow \mathbb{R}$ defined by $F(u) = max_{t \in [a,b]} \lvert u(t) \rvert$. Can you find a function $h \...
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How to compute $D(g \circ f)_x$ in Banach spaces?
Consider Banach spaces $X, Y_i, Z$ and let $U \subseteq X$, $V \subseteq Y_1 \times \ldots \times Y_n$ and let $f:= (f_1, \ldots, f_n): U \rightarrow V$ and $g: V \rightarrow Z$.
Assume that $f$ is ...
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Frechet differential in $l^\infty$
Let $(f_n)_{n \in \mathbb{N}}$ with $f_n \in \mathcal{C}^1(\mathbb{R})$ for all $n$ and assume that that $(f_n)_n$ and $(f^\prime_n)_n$ are both uniformly bounded and equicontinuous on every $[a,b] \...
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Norm integral inequality for Fréchet-valued functions
On "The inverse function theorem of Nash and Moser" paper by Richard S. Hamilton he discusses the Calculus theory for continuous function $f:[a,b]\longrightarrow F$, where $F$ is a Fréchet ...
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Pathwise differentiation under integrated measurement (stochastic volatility) models.
Setup
Consider the (general) stochastic volatility model
\begin{align}
\mathrm{d}X_t &= \mu^X(X_t)\mathrm{d}t + \sigma^X(X_t)\mathrm{d}W^X_t \\
\mathrm{d}Y_t &= \mu^Y(X_t)\mathrm{d}t + ...
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Frechet derivative of Integral operator
In the following let $g(x,r,u)$ be a continuous function $g:[a,b]\times \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$. Furthermore we assume that the partial derivative $\frac{\partial g}{\...
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Differentiation and continuity
I would like to find a proof on the following proposition.
Let $X$ and $Y$ be two Banach spaces and $O$ an open set of $X$ and $\overset{\sim}{X}$ be a dense subspace e of $X$. Let $F : O \to Y$ be a ...
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If $F$ has bounded linear Gateaux derivative at $x_0$, what condition guarantees a Fréchet derivative?
Let $X$ and $Y$ be real normed spaces and suppose $F:X\to Y$ has a bounded linear Gateaux derivative $F'$ at $x_0\in X$. (In the following, $F'$ also denotes a Gateaux derivative at other points in $X$...
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Limit of Fréchet differentiable functions
If I have a sequence of Fréchet differentiable functions, under what conditions can I prove that the limit is Fréchet differentiable?
For example, suppose that I have $f:\ell^p(\mathbb{R})\to\mathbb{R}...
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Extend the construction of derivative
I'm reading about derivative in Amann's Analysis I. The authors define derivative for functions defined on arbitrary subset of $\mathbb K \in \{\mathbb R, \mathbb C\}$. To ensure that the operation of ...
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Flaw in proof of Jacobi's formula on Wikipedia?
The Wikipedia article on Jacobi's formula (which gives the differential of the determinant function) contains two proofs, the second of which begins with a lemma claiming
$\det'(I)=\operatorname{tr}$,...
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Second Derivative Test in Banach Spaces
According to $[$Exercise $12.8$, $1]$ we have the following version of the Second Derivative Test:
Theorem. Let $E=(E,\|\cdot \|)$ be a Banach space, let $D$ be a subset of $E$ and suppose $f: D \...
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Does this version of inverse function theorem hold for Banach space?
Let $E, F$ be Banach spaces over $\mathbb{K} \in \{\mathbb{C}, \mathbb{R}\}$. Let $\mathcal L_{\text{is}} (E, F)$ be the set of all topological isomorphisms from $E$ to $F$. Then $\mathcal L_{\text{is}...
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Given a derivative, say, as a tensor, is the corresponding linear approximation unique?
I try to compute the linear approximation to, say, $f(A)=A^{-1}$ using a result such as http://www.matrixcalculus.org/ that returns $-A^{-1}\otimes A^{-1}$ which is not a linear function but a matrix (...
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Relatio between Frechet derivative and partial derivatives in $\mathbb R^n$
Let $X$ Bananch space and $f \colon X \to \mathbb R$, $A$ open in $X$. $f\in \mathcal C^1(A)$ if it Frechet differentiable for all $x \in A$ and its Frechet derivative $Df \colon X \to \mathcal L (X, \...
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Is this a Gateaux differential?
Given an Hilbert space $H$ P.L.Lions in [Lions, Pierre-Louis. "Viscosity solutions of fully nonlinear second-order equations and optimal stochastic control in infinite dimensions. Part I: The ...
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Composition between Frechet and Gateaux derivative
I am currently dealing with the two terms "Gateaux-derivation" and "Frechet-derivation" and would like to know if there is a Frechet differentiable function $f$ and Gateaux ...
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Computation of the derivative of a matrix-valued function
Let $a,b,c \in \mathbb{N}$, $f:\mathbb{R}^a \rightarrow \mathbb{R}^{b \times c}$ be Frèchet-differentiable with Frèchet-derivative $Df$ and let $y,x \in \mathbb{R}^a$.
Is there a way to compute $(Df)(...
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Explanation of Optimal Control Theory in terms of Frèchet derivatives
I'm interested in getting a better understanding of optimal control theory including the HJB equation and the Maximum principle. Most of the books/notes I've seen do not seem to use a normed space ...
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Equivalence of Frechet derivative to first order expansion
Let $V$ and $W$ be normed vector spaces, and $U \subset V$ be an open subset of $V$. A function $f: U \to W$ is called Frechet differentiable at $x \in U$ if there exists a bounded linear operator $A: ...
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Differentiation along directions for maps on $\ell^2(\mathbb{N})$
Let $\ell^2 \equiv \ell^2(\mathbb{N})$ denote the space of square-summable real sequences.
Let $f \colon \ell^2 \to \ell^2$. Let $f_v(x) := \langle v, f(x) \rangle$.
Suppose there exists a continuous ...
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An open, limited and connected set $C$ under the hipotesis of the theorem of inverse function with $f(\partial C)\cap C = \emptyset$
PROBLEM:
Consider $(V,||·||_{V})$ Banach, $U\subset V$ open and $f:U\rightarrow V$ of class $C^{1}$ with Frechet derivative invertible in all $U$. Supose that $A\subset U$ is open, connected and $\bar{...
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Constrained optimization with constraint qualification conditions
Let $X$ be a normed linear space, and let $G$ be a Frechet differentiable mapping defined on $X$.
A point $x_0$ is said to satisfy the constraint qualification relative to the inequality $G(x)\leq\...
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Derivative of a multivariable function defined with the inner product
Let $n\geq 1$ and let $f: \mathbb{R}^n \to \mathbb{R}^n$ be a $C^1$ function. We know that for a fixed $x=(x_1,....,x_n)$ in $\mathbb{R}^n$, the derivative of $f$ at $x$, denoted by $f'(x)$, is a ...
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Gâteaux differentiable function (with linear continuous Gateaux derivative) that is not Fréchet differentiable
This is my first ever post on math stack exchange, so please forgive me if make any beginner mistakes and thank you for your help :)
I am an undergraduate math student doing a summer reading program ...
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$f'$ has a limit $\Longrightarrow$ $f$ has a limit without codomain Banach?
Using Hahn Banach and Cauchy criterion I am able to prove the following:
Let $X$ normed, $Y$ Banach and $U \subset X$ open. Suppose that for $x_0 \in U$ we have $f : U \setminus \{x_0 \} \...
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How to bound non-linear terms of integral operator in Sobolev Space
Let $I=[0,1]$ and consider the function $F:H^1(I)\to \mathbb R$ given by $$F(u)=\int_I \bigr( u(t) \bigr)^4 dt. $$
This makes sense, because $H^1(I)\hookrightarrow C^0(I).$ I want to understand the ...
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Question about Frechet differentiability.
Consider $F(f) = f^2$ with $F:L^2[0,1] \to L^1[0,1]$. We want to know if it's differentiable by Frechet.
So based on definition:
$$
\underset{\|h\|_{L_2} \to 0}{\lim}\dfrac{\|(f+h)^2 - f^2 - A(f) h\|_{...
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Is there a sense in which a function converges to its total derivative?
In short: Is there a way to rigorously define the total derivative (of $F$ at $x$) as a function $dF_x$ which is (1) linear and (2) a usual topological limit of some function $\mu:X\to Y$, where $X$ ...
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A lipschitz function $f(x)$ valued in the unit ball has almost everywhere Frechet derivative orthogonal to $f(x)$ if $\|x\|=1$.
Consider a function $f:R^n\to R^m$ valued in the unit ball $B=\{u\in R^m: \|u\|=1\}$.
Assume $f$ is Lipschitz. By Rademacher's theorem, $f$ is differentiable almost everywhere, i.e., for almost every $...
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Reference for chain rule for continuously Frechet differentiable maps
Let $f\colon L^p(\Omega;X) \to L^q(\Omega;Y)$ and $g\colon L^q(\Omega;Y) \to L^r(\Omega;Z)$ where $X,Y,Z$ are separable Hilbert spaces and $\Omega$ is a smooth and bounded open set (eg. $\Omega = [0,T]...
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Chain rule and generalized composition of multilinear maps
I know that if functions $f : \mathbb{R}^n \to \mathbb{R}^m$ and $g : \mathbb{R}^m \to \mathbb{R}^p$ are differentiable at $x \in \mathbb{R}^n$ and $f(x) \in \mathbb{R}^m$, respectively, with ...
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Derivative of a function from a closed Riemannian manifold to a Hilbert space
Let $F$ be a Hilbert space and $\Theta$ a $d$-dimensional closed Riemannian manifold.
Consider the twice Fréchet differentiable functions $R \colon F \to [0, \infty)$ and $\phi \colon \Theta \to F$ ...
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