# 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, perturbation by flow

I have read this version of the definition of functional derivative https://en.wikipedia.org/wiki/Functional_derivative#Functional_derivative In the above definition the functional derivative takes ...
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### Fréchet derivative of a (nonlinear) differential operator

The question may not be too well-posed, but loosely speaking, suppose $L:W^{1,p}(\mathbb R)\to L^p(\mathbb R)$ is a (possibly nonlinear) first order differential differential operator, such that all ...
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### If $f \in W^{1, 1}(\mathbb{R}^n)$ then $\theta \mapsto f \circ (Id + \theta)$ is differentiable

In a book I'm reading I have this statement: If $f \in W^{1, 1}(\mathbb{R}^N)$, then the map defined by \begin{align} W^{1, \infty}(\mathbb{R}^N; \mathbb{R}^N) &\rightarrow L^1(\mathbb{R}^N)\\ \...
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### Chain rule involving Fréchet derivative

Suppose that $P(w)$ is a probability density function with support $w \in [0,\infty)$ and $G = G[P]$ is a functional satisfying $G[P] \in [0,1]$. I saw a paper used a chain rule of the following form: ...
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### Fréchet derivative of scalarfield operator $f \mapsto \nabla f \cdot \sigma$?

Let $\sigma \in \mathfrak{X}(\Omega)$ be a fixed smooth vector field on an open domain $\Omega \subset \mathbb{R}^d$. Consider the operator $T\colon C^\infty(\Omega) \to C^\infty(\Omega)$ on the space ...
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### Determine the tangent space of the boundary of a manifold of the form $f(\mathbb R^d)$ for a differentiable $f$

Let $d\in\mathbb N$; $x^\ast\in\mathbb R^d$; $k\in\mathbb N$; $f:\mathbb R^d\to\mathbb R^k$ be Fréchet differentiable at $x^\ast$; $v^\ast:=f(x^\ast)$; $A:={\rm D}f(x)$; $M:=f(\mathbb R^d)$. Assume ...
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### Best linear approximation in a neighborhood property for Frechet derivative

Let $D$ denote a Frechet derivative at a point $a$. Then for $f:\mathbb{R}\to\mathbb{R}$ it holds for all $x$ from some neighborhood of $a$: $$|f(x)-(f(a)+D(x-a))|\leq|f(x)-(f(a)+M(x-a))|,\ M\neq D$$ ...
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### what is the difference between Strong Fréchet derivative and Fréchet derivative

I am confused about the relationship between Frechet differentiable and strong Fréchet differentiable. Assume the function $f(x) \in \mathbb{R}, x\in \mathbb{R}^n$ is Fréchet differentiable at $x$. ...
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### If $f$ is an isometry between open sets, is $\left|\det{\rm D}f\right|=1$?

Let $d\in\mathbb N$ and $\Omega_i\subseteq\mathbb R^d$ be open. If $f$ is an isometry from $\Omega_1$ to $\Omega_2$, can we conclude that $f$ is Fréchet differentiable and $\left|\det{\rm D}f\right|=1$...
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### Difficulty in proving that $\Phi$ is differentiable.

I am studying Lie algebra and Lie groups from the lecture notes given by our instructor. Here I find a definition of differentiable Lie algebra homomorphism. What it says is as follows $:$ Let $E$ be ...
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### How to prove that $\phi$ is continuous?

Definition $:$ Let $E$ be a Banach space. Let $G,H \subseteq GL(E)$ be linear Banach Lie groups (i.e. they are closed subgroups of $GL (E)$). A group homomorphism $\Phi : G \longrightarrow H$ is said ...
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### Local integrability of a Lebesgue integral

Fix a Banach space $X$, an open subset $U \subset X$, and a measure space $(\Omega, \mathcal A, \mu)$. Let $f: U \times \Omega \to \mathcal R$ be continuously Frechet-differentiable in the first ...
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### Boundedness of the first Fréchet derivative

Let $(E,\Vert\cdot\Vert)$ be a normed vector space and $f:E \rightarrow \mathbb{R}$ be a twice Fréchet differentiable function with $\sup_{x \in E} \frac{\vert f(x)\vert}{1+\Vert x\Vert^3} < \infty$...
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### Two versions of Fréchet derivatives in terms of Landau notation

In the wiki article, $F$ is Fréchet differentiable at $x$ if there exists a function $A$ that is linear in $h$ such that $F(x+h)-F(x)-Ah=R(h)$ is $o(h)$, meaning that for every $\epsilon> 0$, there ...
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### Does Frechet differentiability of the first order partials imply twice Frechet differentiability?

Let $f:\mathbb{R}^n \to \mathbb{R}$. If all first order partial derivatives of $f$ are Frechet differentiable, must $f$ be twice Frechet differentiable? This question is inspired by my comment on this ...
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### How to compute Frechet Derivatives that involve matrices.

Suppose I have a linear operator which maps $S^n\to\mathbb R^{n \times n}$. It is represented as $LX:= XA^T + AX.$ How can I find the Frechet derivative of $f(X) = XA^T + AX.$