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Questions tagged [gateaux-derivative]

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Deriving the Euler-Lagrange Equation using the Gateaux Derivative

Can anyone explain how the professor goes from line 4 to 5 of the derivation? In particular, how is: $$\frac{\delta L}{\delta u}h'=-\frac{d}{dx}\frac{\delta L}{\delta u'}h$$ The professor states ...
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43 views

Frechet Derivative of a Function Containing an Integral Operator Applied to the Differentiated Term

I'm new to Frechet and Gateaux differentiation, this likely has some mistakes. I was hoping someone could check this and offer some guidance. Suppose we have the following functional: \begin{equation}...
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1answer
27 views

Can a variational inequality (necessary condition for minima) be strict?

A well-known fact in optimization theory is the following: Let $C_{ad}$ be a non-empty, convex subspace of a real Banach space $B$ and let $F: U \mapsto \mathbb{R}$ be a function defined on an open ...
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Gateaux Derivative of Fourier Transform/Characteristic Function

Suppose that $X$ is a square integrable random-variable defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. It's characteristic function/Fourier transform is defined to be $$ \mathfrak{...
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1answer
46 views

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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1answer
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Necessity of the Hahn Banach Theorem for the Gateaux Mean Value Theorem

The following theorem is in Drabek, Milota's Nonlinear Analysis. Like Drabek and Milota, I won't assume a priori Gateaux differentials are continuous nor linear. Theorem. Let $X,Y$ be normed spaces, ...
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1answer
78 views

Taylor expansion for Gâteaux derivative

Let $\mathbb{X}$ be a normed Space and $ f: \mathbb{X}\mapsto\mathbb{R} $ is twice Gâteaux differentiable (not necessary Fréchet differentiable). Is it possible to build a Taylorexpansion for $f$ in ...
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Question regarding Gateaux differentiability

Let $E$ be a normed space and $\Omega \subset E$ be an open convex subset. Let $a\in \Omega$ and $f:\Omega\longrightarrow \mathbb{R}$, we say $f$ is differentiable in the direction $v$ at $a$ if the ...
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34 views

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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prove that the functional is $\alpha$-elliptic

I got a nonlinear functional who is convex and Gâteaux differentiable. Is there some property of these two that can bring me that the functional is $\alpha$-elliptic???
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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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First Variation of CDF inside an Indicator Function

I would like to minimize the functional $\mathcal{F}(\mu) = \int x I(F_\mu(x)\leq\tau) d\mu(x)$. However, I'm don't understand how to find the first variation of the term $I(F_\mu(x)\leq\tau) = I(\mu((...
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Hadamard Derivative of time transformed stochastic process

Given a continuous time stochastic process $X(t)$, we can define the functional transformation, $$f(X)(t) = (X(t))^2 - 2X(t)$$ and evaluate the Hadamard derivative. Given a transformation on the real ...
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Hadamard differentiability of map implies remainder term is zero at a point?

Let $f\colon X \to Y$ be Hadamard differentiable. This implies that $$f(x+th)-f(x) = tf'(x)(h) + o(t)$$ holds where $o$ is a remainder term, i.e. $t^{-1}o(t) \to 0$ as $t \to 0$. Does it follow that ...
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1answer
146 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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Differentiability of $L^p$-valued functions

Let $(\Omega,\mathcal A,\mu)$ be a finite measure space $X$ be a metric $\mathbb R$-vector space $h\in X$ $\Lambda\subseteq X$ be open $Y$ be a $\mathbb R$-Banach space $p\ge1$ If $f:\Omega\times\...
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Find a counterexample that $f(x)$ is Gateaux differentiable and $\lambda(x)$ is not continous [duplicate]

I've learned that function $f:\mathbb{R}^n\rightarrow \mathbb{R}^p$, at $x \in \mathbb {R}^n$, $f(x)$ is Gateaux differentiable, then exits a linear operator $\lambda (x):\mathbb{R}^n\rightarrow \...
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Differentiability of Norms of $l_{\infty}$

In the book Fabian and others I saw exercise: "Let $\|$.$\|_{\infty}$ denote the canonical of $l_{\infty}$ and set $p(x) = \limsup |x_i|$. Define $\||x\|| = \|x\|_{\infty} + p(x)$ for $x \in l_{\...
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1answer
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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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1answer
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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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1answer
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An example of Gateaux derivative

In one of the Gateaux derivative examples mentioned here on page 5, it is mentioned that the function $f(x) = \lvert x\rvert $ has the Gateaux derivative defined at $x=0.$ In particular, it says that ...
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1answer
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Why does “t” in the formula for gateaux differentiation represent a scalar value?

$$\lim \limits_{t \to 0} \frac{f(x_0+tv)-f(x_0)}{t}$$ This is the formula I have in my textbook for a function defined in $\mathbb{R}^p$ with values in $\mathbb{R}^q$ that calculates the directional ...
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Counter-example for Gateaux derivative being continuous

Let $X$ and $Y$ be infinite-dimensional Banach spaces over $\mathbb{C}$ and $U\subset X$ be norm open. We say $f:U\rightarrow Y$ is Gateaux differentiable if for each $x\in U$ and each $h\in X$, $$Df(...
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Failure of chain rule for Gâteaux derivative on an arbitrary Banach space

Let $X,Y$ be Banach spaces. A function $f :X \to Y$ is said to be Gâteaux differentiable at $x$ if there exists a bounded linear operator $A : X \to Y$ such that $$\lim_{r \to 0}\frac{\|f(x+rh)-f(x)-...
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0answers
58 views

Proving/disproving a sufficient condition for Gâteaux differentiability

Let $X,Y$ be Banach spaces. A function $f :X \to Y$ is said to be Gâteaux differentiable at $x$ if there exists a bounded linear operator $A : X \to Y$ such that $$\lim_{r \to 0}\frac{\|f(x+rh)-f(x)-...
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2answers
199 views

Meaning of functional differentiability

I just began studying variational calculus, and I'm having some issues getting a conceptual grasp on functional differentiability. Let $J[y]$ be a functional defined on some normed linear space, ...
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1answer
293 views

Frechet and Gateaux derivative of integrals

I am new to differential calculus on normed spaces and I struggle with some easy things. Let $f:[a,b]\times\mathbb{R}\longrightarrow\mathbb{R}$ and $g:\mathbb{R}\longrightarrow\mathbb{R}$ two ...
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1answer
110 views

Is my derivation of the Gateaux derivative correct and rigorous?

The textbook on calculus of variations by Liberson gives the following definition of "first variation": It also gives the definition of the "Gateaux derivative" I want to prove that if $G$ is the ...
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1answer
45 views

Directional derivative of $f:H^1 \to H^1$

Let $f:\mathbb{R} \to \mathbb{R}$ be a smooth function. We think of $f:H^1(\Omega) \to H^1(\Omega)$ in the sense that $f(u)(x) = f(u(x))$. I want to know what the directional derivative is of $f:H^1(\...
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1answer
224 views

Confusion about linearity of Gateaux Derivative

I have the following formulation of the Gateaux derivative for functions $f:\mathbb{R}^m\to\mathbb{R}^n$. Let $f(x) = \sum_{i=1}^n f_i(x)e_i$ where $e_i$ forms a basis for $\mathbb{R}^n$, and $f_i:\...
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1answer
201 views

Don't follow proof that Hadamard differentiable implies compactly differentiable

A function $f:X \to Y$ between reflexive separable Banach spaces is said to be compactly differentiable if $$\lim_{t \to 0} \frac{f(x+th) - f(x) }{t} -f'(x)(h) =0$$ where the limit holds uniformly in ...
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1answer
358 views

Gateaux and Frechet derivatives on vector valued functions

If $f:\mathbb{R}\to\mathbb{X}$ is a function from the real numbers to any normed vector space (finite or infinite dimension), and $f$ is Gateaux differentiable, is $f$ necessarily Frechet ...
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122 views

Different definitions of the continuity of the Gâteaux-Derivative are inequivalent in Banach spaces

I am wondering about a problem from the calculus of derivatives in Banachspaces. It is about the difference between two definitions of continuity concerning the Gâteaux-Derivative of a function $P:X\...
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1answer
151 views

Directional derivative of risk measure

Let $S,Z \in L^\infty(\Omega,\mathcal{F},\mathbb{P})$ and $\rho \colon L^\infty \rightarrow \mathbb{R}$ be a risk measure. When is the directional derivative (Gâteaux derivative) $$\lim_{\varepsilon \...
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1answer
155 views

What is the infinite dimensional counterpart of the Lie derivative?

In a finite dimensional space, one calculates the Lie derivative as $L_f(g)(x) = \langle \nabla g, f \rangle$ What is the equivalent in an infinite dimensional space? For example if $g$ takes as ...
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2answers
667 views

Jacobian matrix and Gateaux derivative

Let $f:\mathbb{R}^N\rightarrow\mathbb{R}^M$ be a function which is Gâteaux differentiable and let $J_f\in\mathbb{R}^{M\times N}$ be its Jacobian matrix. Is it true that the Gâteaux derivative of $f$ ...
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1answer
229 views

Is this function Gateaux differentiable?

Consider the real-valued function $f:\mathbb{R}\rightarrow\mathbb{R}$ of two real variables defined by $$f(x,y)=\begin{cases} \frac{x^3}{x^2+y^2} & \mbox{ if } (x, y)\ne (0, 0)\\ 0 & \mbox{ if ...
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1answer
32 views

$F$ with Gateaux-derivative $A$, then $pF(u)=A(u)(u)$

Let $F:X\to \mathbb{R}$ Gateaux-differentiable with Gateaux-derivative $A:X\to X^*,$ ($X^*$) is the dual space pf $X$. Let $p\in\mathbb{R}$ such that $$F(\lambda u)=\lambda^pF(u)$$for all $u\in X$ and ...
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1answer
31 views

well defined $F(u)=\int_I \psi udx$ and it's Gateaux derivative is $DF(u)=\psi(u)$

Let $1\le p <\infty$, $\psi\in C^1(\mathbb{R})$ such that $\psi (0)=0$ and $|\psi '(y)|\le C|y|^{p-1}$ for all $y\in\mathbb{R}$ and for a constant $C>0$. Prove that $F:L^p(I)\to \mathbb{R}$ ...
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1answer
607 views

Show that $G(y^*)(h) = \int \limits _X {y^* \circ F(t) \Bbb d \nu _h (t)}$ for some measure $\nu$ on $X$

The following image is taken from this paper, page $129$. Questions: Why do we have $$G(y^*)(h) = \int \limits _X {y^* \circ F(t) \Bbb d \nu _h (t)} ?$$ (I couldn't get my hands on the book '...
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0answers
28 views

existence directional derivatives

I've this function : $$ f(x,y)= \begin{cases} \dfrac{(1+x^2)x^2y^4}{x^4+2x^2y^4+y^8} \quad \text{ for } \qquad (x,y)\ne (0,0) \\ 0 \quad \quad \quad \quad \quad \quad \quad \quad \text{ ...
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2answers
51 views

Is $ \left| \lim_{t \rightarrow 0}{\frac{f(x+ty) - f(x)}{t}} \right| \leq \lim_{t \rightarrow 0}{ \frac{|f(x+ty) - f(x)|}{|t|}}$

Suppose $X$ is a Banach space and $f:X \rightarrow \mathbb{R}$ is a continuous function. Is it true for all $x,y \in X$ that $$ \left| \lim_{t \rightarrow 0}{\dfrac{f(x+ty) - f(x)}{t}} \right| \leq \...
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1answer
67 views

Show that $f=1/2 ||T(x)||^2$ is Gateaux differentiable everywhere.

Let $T$ be a bounded linear transformation from a real Hilbert space $H$ onto itself and define $f$ by $$f(x)=\dfrac{1}{2}||T(x)||^2.$$ How can I show that $f$ is Gateaux differentiable everywhere? ...
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0answers
52 views

Definition of $\nabla f(ux_n - S(t))$ on a Banach space

Suppose $X$ is a Banach space and $f$ is a Lipschitz Gateaux differentiable function on $X$. Let $S : [0,1]^{\mathbb{N} \backslash \{ n \}} \rightarrow X$. Then we have $$f(x_n + S(t)) - f(S(t)) = ...
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1answer
68 views

show that $(y^* \circ F) * \gamma(x) = \int_X {y^* \circ F(x+t)}d\gamma(t)$ is Gateaux differentiable everywhere

Suppose $X$ is a separable Banach space and $Y$ is a Banach space. If $F:X \rightarrow Y$ is a Lipschitz map and $\gamma$ is a nondegenerate Gaussian probability measure on $X$ with mean $0$, ...
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1answer
105 views

Show that a function is nowhere Gateaux differentiable.

Suppose $X$ is a Banach space. For any $x \in X$, define the set $\mathcal{F}(X) = \overline{span \{ \delta_x : x \in X \}}$ where $\delta_x(f)=f(x)$ for all $f \in$ $Lip_0(X)$. The set Lip$_0(X)$ ...
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1answer
224 views

Riesz isomorphism and Gateaux derivatives

I'm reading a paper and I wonder about the statement: $f'(x)h=(y(x),h)_{L^2}$ then follows with the Riesz isomorphism that $f'(x)=y(x)$ $f'(x)h$ is the Gateaux derivative. I don't see that with the ...
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0answers
42 views

Find a condition for being the $f$ Gateaux differentiable at $0$, and that, consequently,the derivative of Gateaux is not necesarily linear map

Show that the derivative of Gateaux exists, and is the linear map, if $f$ is differentiable with the ordinary meaning.If $f:R^n \to R$ with $$f(x)={g(x)\over h(x)}$$ where $g,h \in C^{\infty}(R^n)$,...
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2answers
1k views

What is an example of Gâteaux differentiable but not Fréchet differentiable at a point in a finite-dimensional space?

Let $V,W$ be nonzero normed spaces over $\mathbb{K}$ such that $V$ is finite-dimensional. Let $E$ open in $\mathbb{K}$ and $p\in E$. Let $f:E\rightarrow W$ be Gâteaux-differentiable at $p$. Is $f$ ...
3
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1answer
610 views

Prove that continuity in x of the Gateaux derivative, $f'(x;y)$, implies Frechet differentiability

Prove that continuity in x of the Gateaux derivative implies Frechet differentiability Let $x$ be te point, $y$ the direction and $f'(x;y)=y·a(x)$. First, I considere the function $g(\varepsilon)=f(...