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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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:
$$
\underset{\|h\|_{L_2} \to 0}{\lim}\dfrac{\|(f+h)^2 - f^2 - A(f) h\|_{...
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Is there a sense in which a function converges to its total derivative?
In short: Is there a way to rigorously define the total derivative (of $F$ at $x$) as a function $dF_x$ which is (1) linear and (2) a usual topological limit of some function $\mu:X\to Y$, where $X$ ...
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A lipschitz function $f(x)$ valued in the unit ball has almost everywhere Frechet derivative orthogonal to $f(x)$ if $\|x\|=1$.
Consider a function $f:R^n\to R^m$ valued in the unit ball $B=\{u\in R^m: \|u\|=1\}$.
Assume $f$ is Lipschitz. By Rademacher's theorem, $f$ is differentiable almost everywhere, i.e., for almost every $...
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Reference for chain rule for continuously Frechet differentiable maps
Let $f\colon L^p(\Omega;X) \to L^q(\Omega;Y)$ and $g\colon L^q(\Omega;Y) \to L^r(\Omega;Z)$ where $X,Y,Z$ are separable Hilbert spaces and $\Omega$ is a smooth and bounded open set (eg. $\Omega = [0,T]...
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Chain rule and generalized composition of multilinear maps
I know that if functions $f : \mathbb{R}^n \to \mathbb{R}^m$ and $g : \mathbb{R}^m \to \mathbb{R}^p$ are differentiable at $x \in \mathbb{R}^n$ and $f(x) \in \mathbb{R}^m$, respectively, with ...
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Derivative of a function from a closed Riemannian manifold to a Hilbert space
Let $F$ be a Hilbert space and $\Theta$ a $d$-dimensional closed Riemannian manifold.
Consider the twice Fréchet differentiable functions $R \colon F \to [0, \infty)$ and $\phi \colon \Theta \to F$ ...
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Solution of ODE Gâteaux-differentiable w.r.t. RHS?
Assume we have an atonomous ODE where $f \in C^1(\mathbb{R})$
$$
\dot{x}(t) = f(\dot{x}(t)), ~x(0) = x_0
$$
with some unique solution $x_f$ that of course depends on $f$.
I am courious whether $f \...
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Fréchet derivative of the total variation norm for measures on a manifold
Let $\Theta$ be a compact $d$-dimensional Riemannian manifold without boundary and $M(\Theta)$ (resp. $M_+(\Theta)$) denote the set of signed (resp. nonnegative) finite Borel measures on $\Theta$.
...
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Application of the Frechet derivative
$f\colon U\subset \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$ is differenciable at $x_{0}$ if there exist a linear transformation $T\colon \mathbb{R}^{n}\longrightarrow\mathbb{R}^{m}$, such that:
\...
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Find linear transformation in function space
Given $f$ and $g$ functions in a ball $B$ of the $L^p([0,1])$ space and the operators $F_i:B\times B\rightarrow \mathbb{R}$ such that
$$\sum_{i=1}^\infty k_i F_i[f,g]=T_f\cdot g$$ where $k_i\in \...
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Range at two points of the differential of a surjective map are equal?
Let $\mathcal{H}_1$ and $\mathcal{H}_2$ be two Hilbert spaces such that $\dim\mathcal{H}_2<\infty$, and let $F:\mathcal{H }_1\rightarrow\mathcal{H}_2$ be a Fréchet differentiable surjective map. ...
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Fréchet derivative of $ \varphi:\mathbb R^q \to \mathbb R, g \mapsto \sum_{i=1}^n \langle g-b_i, a_i\rangle^2$
Fix $a_i, b_i \in \mathbb R^q$ for $i=1, \ldots, n$. Consider the map
$$
\varphi:\mathbb R^q \to \mathbb R, g \mapsto \sum_{i=1}^n \langle g-b_i, a_i\rangle^2.
$$
I would like to compute the Fréchet ...
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Why is this functional derivative equal to $0$?
I am currently reading the paper Exponential Convergence Rates in Classification (2005) by Vladimir Koltchinskii and Oleksandra Beznosova, and I'm having trouble following the proof of the main result,...
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Deducing linear conjugacy from conjugacy of linear systems
Given $X, Y \in \mathbb{R}^{n \times n}$ and a $C^1$ surjective diffeomorphism $F: \mathbb{R}^n \to \mathbb{R}^n$, suppose that $e^{tX} \circ F = F \circ e^{tY}$ for all real $t \in [0, \infty)$. I ...
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How to obtain integral remainder in Taylor's formula for functions between Banach spaces?
In Amann's textbook Analysis II, the authors present below version of Taylor's theorem.
Let $E,F$ be Banach spaces. Suppose $X$ is open in $E, q \in \mathbb{N}^{\times}$, and $f$ belongs to $C^{q}(X,...
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How do higher-order Fréchet derivatives look like in case of $f:\mathbb R \to \mathbb R, x \mapsto x^3$?
Recently, I have come across the Taylor's formula for functions between Banach spaces. To better understand the machinery, I try to see how higher-order derivatives look like in a simple example, i.e.,...
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Is the Integral Operator $L_p(\mathbb{R}^n)\ni\varphi\mapsto \int_{\mathbb{R}^n} \varphi (x) g(x) dx, g\in L_{p'}(\mathbb{R}^n)$ (Frechet-) Smooth?
I don't know much about Frechet-differentiability. I just need to know if the following statement is true.
Let $p^{-1}+p'^{-1}=1$. Let $g\in L_{p'}(\mathbb{R}^n,\mathbb{R})$. Then the operator
$ G: ...
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Chain rule for differentiation yields conflicting dimensions
Suppose I have differentiable functions (in the sense of the frechet derivative)
$f\colon \mathbb{R} \to \mathbb{R}^{n\times n} $ and $g\colon \mathbb{R}^{n} \to \mathbb{R}$
, where $f$ is a linear ...
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How can we show that $x(t)-y(t)=\int_{s-ε}^tf(x(r),a)-f(y(r),a(r))\:{\rm d}r$ implies $x(s)-y(s)=(f(x(s),a)-f(x(s),\alpha(s)))\varepsilon+o(ε)$?
Let $E_i$ be a $\mathbb R$-Banach space, $x,y:[0,T]\to E_1$ be continuous, $a\in E_2$, $\alpha:[0,T]\to E_2$ be bounded and Borel measurable and $f:E_1\times E_2\to\mathbb R$ be Fréchet differentiable ...
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For which class $H$ of functions is $H\ni x\mapsto\int_0^Tf(t,x(t))\:{\rm d}t$ differentiable?
Let $T>0$, $X,Y$ be Banach spaces, $f:[0,T]\times X\to Y$ be Fréchet differentiable in the second argument and such that ${\rm D}_2f$ is (jointly) continuous, $H$ be a subspace of all Borel ...
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Symmetry of second derivatives: Can we conclude $({\rm D}_2({\rm D}_1f)(x)h_2)h_1'=({\rm D}_1({\rm D}_2f)(x)h_1)h_2'$?
Let $E_i$ be a normed $\mathbb R$-vector space.
From Schwarz's theorem we know that if $\Omega\subseteq E_1$ is open and $f:\Omega\to E_2$ is twice differentiable at $x\in\Omega$, then ${\rm D}^2f(x)$ ...
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How to express the Frechet derivative in terms of a gradient vector.
Is it possible to express the Frechet derivative in terms of gradient vectors in normed vector spaces? Since the definition tells us the following,
Let $V$ and $W$ be normed vector spaces, and $U\...
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Help proving that the derivative $Df(x)$ is continuous
In what follows, $X, Y$ are Banach and $U \subseteq X$ is open. Consider the map $f: U \to Y$. We say that $f$ is locally uniformly differentiable at $x \in U$ and $h \in X$ if given $\epsilon > 0$ ...
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How to set up a constraint optimization problem in quadratig equations in banach spaces?
I am searching for details to find how to set up a constraint optimization problem in quadratic equations in banach spaces. In particular I search for details about the cases where Frechet derivative ...
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When an ordinary partial derivative is a Frechet derivative on a Banach space?
We have a function $p(x, \theta)\in \mathbb{R},\ x\in\mathbb{R},\ \theta\in\mathbb{R}^n$.
We can also consider $p(\cdot, \theta)$ to be a map from $\mathbb{R}^n$ to a Banach space of functions. For ...
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Apply the mean value theorem to find $x_0$ with $\varphi(g(b))-\varphi(g(a))=\varphi'(g(x_0))(g(b)-g(a))$
Let $\varphi:\Bbb R\to\Bbb R$ be continuously differentiable and $g:[a,b]\to\Bbb R$ be continuous. Are we able to show that $$\varphi(g(b))-\varphi(g(a))=\varphi'(g(x_0))(g(b)-g(a))\tag1$$ for some $...
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Can we show $\varphi(g(b))-\varphi(g(a))=\int_a^b{\rm D}\varphi(g(t))\:{\rm d}g(t)$?
Let $E,F$ be $\mathbb R$-Banach spaces, $\varphi:E\to F$ be (continuously) Fréchet differentiable, $a,b\in\mathbb R$ with $a<b$ and $g:[a,b]\to E$ be continuous and of bounded variaton.
Are we ...
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Converse to Taylor's theorem in Banach spaces
So I have been following a book and understand the following version of Taylor's theorem:
Let $X$ be a Banach space and $Y$ a separable Banach space, $A \subset X$ open and convex, $f\in C^n(A, Y)$ ...
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Is this Fréchet derivative really a bounded functional?
I am currently studying Newton's method on Banach spaces, specifically as it pertains to the linearization of nonlinear PDEs. In a sample problem, the following operator between Banach spaces came up:
...
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Where is the $\Vert \cdot \Vert_{\infty}$ norm Fréchet differentiable on $c_0$?
At what points $x\neq 0$ is the mapping $x\mapsto \Vert x\Vert_{\infty}$ Fréchet differentiable on $c_0:=\lbrace (x_n)_{n\in\mathbb{N}} \subset \mathbb{C}:x_n\rightarrow 0\rbrace$? A function $f$ is ...
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Fréchet Derivative of a functional in $L^2$
I need to check if the functional $$ (f(x))(t):=\int_{0}^{1}k(s,t)\cdot \cos(x(t))dt $$ is Fréchet differentiable on $L^2([0,1])$, where $k:[0,1]\times [0,1]\rightarrow \mathbb{R}$ is continuous. A ...
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Prove that $L(f)(x)$ is Fréchet differentiable.
Let $C([0,1])=\{f|f:[0,1]\to \Bbb R$ is continuous$\}$ with norm $||f||_{\infty}=\sup_{x\in[0,1]}|f(x)|$.
Prove that $L:C([0,1])\to C([0,1])$, define by $L(f)(x)=\int_0^x f^2(t)dt$ is Fréchet ...
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What is the Fréchet derivative of a matrix eigenvalue?
I want to determine the Fréchet derivative of eigenvalues over the space of square symmetric matrices $A \in \mathbb{R}^{d \times d}$. I think the result is that if the $p^{\text{th}}$ eigenvalue $\...
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Prove that $DL(x)=L(x)$.
By Fréchet derivative prove that $DL(x)=L(x)$ where $L:\Bbb R^m\to \Bbb R^n$ is a lineal application.
My try:
We have,
$$\begin{align*}\dfrac{L(a+h)-L(a)-DL(a)h}{||h||}&=\dfrac{L(a)+L(h)-L(a)-DL(a)...
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Gateaux derivative and Frechet derivative in calculus of variation
Let $X$ be some Banach space, if $f \in C(U,\Bbb{R})$ a continuous real value function.We define the Gateaux derivative exist if for any $h\in X$ exist $df(x_0,h) \in \Bbb{R}^1$ s.t:
$$|f(x_0+th)-f(...
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How to demonstrate that this infinite pieces piecewise function is Fréchet differentiable?
Say $f: \mathbb{R} \to \mathbb{R}$ is defined as
$
f(x) =
\begin{cases}
sin(2^{-2n}), & |x| = 2^{-n}, n \geq 1, \\
x^2, & \text{otherwise}.
\end{cases}
$
How can we show that this function ...
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functional derivative and dualspace
Consider the function space $F=\{ f : \mathbb{R}^m \rightarrow \mathbb{R}^n\}$ and the empirical scalarproduct:
$$
\langle f,g\rangle:=1/n\sum^n_{i=1}f(x_i)^Tg(x_i),
$$
for a a finite dataset $x_1, \...
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Distributional derivative of a function defined on a Sobolev Space
Let $$J: W^{1,p}(\mathbb{R}) \rightarrow \mathbb{R}$$ be such that $$J(w)=\int_{\mathbb{R}} g(w(t))e^{-at}dt, \quad \mbox{for all} \quad w \in W^{1,p}(\mathbb{R}).$$
Where $$g(s)=\int_0^s \beta(t) dt$$...
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Existence of partial of derivative implies existence of partial of partial
Let $f:\mathbb{R}^m\rightarrow\mathbb{R}^n$ be Frechet differentiable. Show carefully that if $f'$ has directional derivative $D_u f'(a)$ for some $a\in \mathbb{R}^m$ and $u\neq 0$, then for every $v\...
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Analysis II Basic issue applying Frechet chain rule
Let $f$ be a $C^1$ function from $\mathbb{R^n}$ to itself. Let $T=f'(a)$ and $h(x)=T^{-1}( f(x+a)-f(a))$. Then $h'(x)=T^{-1}\circ f'(x+a)$ by the chain rule.
I can derive this result using little $o$ ...
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For a Banach space $E$, is it sufficient to check differentiability of $F: U \to E$ on $\phi \circ F$ for all linear functionals $\phi$?
Let $E$ be a complex Banach space and and $U \subset \mathbb{C}$ an open set containing $0$. Consider a function $F: U \to E$. If for every linear functional $\phi \in E^*$, the function $\phi \circ F:...
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Total Derivative : How to correct the practical definition ? a+h out of set.
Definition : Total Derivative over an open set of a $\mathbb{R}$ vectorial space :
Let E and F be two $\mathbb{R}$ vectorial spaces. Let $U \subset E$ be an open subset of E.
Let $f:U \rightarrow F$ ...
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Fast way to compute Fréchet/Gateaux Derivatives
I was wondering if there is any fast way to compute Fréchet/Gateaux Derivatives, or at least a reasonable guess in most cases, say from the usual derivatives table.
The Fréchet derivatives $\dfrac{\...
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Is one continuous partial derivative and other existing enough to imply diffrentiability of $R^2 \to R$ function
For $f: R^2 \to R$ to be diffrentiable in point $p = (p_x;p_y)$ it has to satisfy for aribitrary $\theta$
$$1)\space\space\lim_{r \to 0} \frac{f(p_x + rcos(\theta);p_y + rsin(\theta)) - f(p_x;p_y) - \...
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confusion between differential, derivative and gradient
Consider a function space $\mathcal{F}$ and functions $F : \mathbb{R}^P \to \mathcal{F}$ and $C : \mathcal{F} \to \mathbb{R}$. Now, the derivative of $C \circ F$ w.r.t. $\theta$ can be written down by ...
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Prove that $A \mapsto A (A + \lambda)^{-1}$ is Frechet differentiable for any positive matrix $A$ and for any $\lambda \gt 0.$
Show that the map $f : A \mapsto A(A + \lambda)^{-1}$ is Frechet differentiable for any positive matrix $A$ and for any $\lambda \gt 0.$ Also find it's Frechet derivative.
Is the space of all ...
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When does separate Frechet differentability imply joint Frechet differentiability?
Let $f\colon X \times Y \to Z$ be a mapping between Banach spaces. If I know that $f(\cdot,y)$ and $f(x,\cdot)$ are Frechet differentiable or $C^1$ functions (for fixed $x$ and $y$), what other ...
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Question about solving Frechet differential equations
I am trying to prove the following fact:
Suppose $\rho$ is a $C^3$ function of real Hilbert spaces $E$ into $F$. Let $\rho''$ denote its second Frechet derivative at some point $x \in E$. We have $\...
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differentiable norm in complex space with inner product
Let $H$ a pre-hilbert complex space (complex space with inner product). Prove that $\|\cdot\|$ and $\|\cdot\|^2$ aren't differentiable (Fréchet) in $H$. I don't know how to use the fact that $H$ is a ...
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