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Questions tagged [gateaux-derivative]

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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, ...
0
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1answer
32 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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25 views

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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17 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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19 views

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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37 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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12 views

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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16 views

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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12 views

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 ...
3
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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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43 views

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_{\...
3
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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
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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96 views

Gâteaux derivative of the supremum norm

Let $u,v \in C[a,b]$ and $M=\{x: \lVert u \rVert_{\infty} = |u(x)| \}$. The Gâteaux derivative of the supremum norm $\lVert \cdot \rVert_{\infty}$ evaluated at $u$ in the direction of $v$ is: $$\...
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1answer
322 views

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
52 views

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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76 views

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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125 views

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
52 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
171 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, ...
4
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1answer
274 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 ...
1
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1answer
93 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 ...
1
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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(\...
2
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1answer
183 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:\...
4
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1answer
147 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 ...
2
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1answer
343 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 ...
3
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0answers
108 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
129 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
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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 ...
3
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2answers
646 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 $f$ ...
2
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1answer
202 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
31 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 ...
0
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1answer
30 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
559 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
27 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
48 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 \...
0
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1answer
61 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)) = ...
0
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1answer
54 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
102 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
196 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
40 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
541 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(...
3
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1answer
606 views

Gateaux derivative of $L_p$ norm

For $2\leq p < \infty$, if we consider $f,g \in L_p(X, \mathcal{M},\mu)$ there is the well-known equality $$\frac{d}{dt}\Vert f+tg \Vert_p^p = \frac{p}{2} \int_X \vert f(x)+tg(x) \vert^{p-2} \left(...
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2answers
1k views

Derivative of the matrix exponential with respect to its matrix argument

I was trying to find the Frechet derivative of $f = \exp(X)$, where $X \in \mathbb{R}^{n\times n}$ is positive definite. I thought it ought to be $\exp(X)$. I see results where the derivative is with ...
4
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1answer
1k views

What exactly is the difference between Gateaux derivative and directional derivative?

The definition of the limit looks very similar between the two derivatives. It seems that directional derivative is the "amount" of the function going in the direction of a vector (arrow), whereas ...
2
votes
3answers
530 views

Locally Lipschitz and Gâteaux Derivative if and only if Frechet Derivative

Consider $f$ locally Lipschitz. So $f$ is Gâteaux Derivative if and only if $f$ is Frechet Derivative. PS.: the converse is trivial.