# 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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### Fréchet Differentiability of a series of norms

I am currently trying to understand the proof of Theorem 9.14 from "Banach Space Theory",Fabian M. et al, which characterizes super-reflexivity. The theorem affirms the following: Let $X$ be ...
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### Uniform Fréchet Differentiability of a series of norms

I'm studying Theorem 9.14 from "Banach Space Theory",Fabian M. et al, which characterizes super-reflexivity. One of the steps of the proof is to show that 2 and 3 imply 4. So, using two ...
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### Uniform Fréchet differentiability

Right now, I'm studying concepts of differentiation in Banach spaces, but I'm pretty new. In several references, I've found the following property: "Let $U\subset X$ be an open convex subset of a ...
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### Characterization of Optimal Payoff (under Expected Utility) via Gateaux-Derivative/Fréchet Derivative

Background: Let $(\Omega, \mathcal{F}, \mathbb{P})$ model a financial market and $T>0$. Denote by $(S_t)_{t\in[0,T]}$ the price process of the risky asset in the financial market. Assume that the ...
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### Derivative of matrix inverse using directional derivative formula

I'm learning about Frechet derivatives of matrices from Bhatia's Matrix Analysis. I want to compute the derivative of the inverse function using the formula for directional derivatives. I've seen on ...
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### Please explain the steps involved after $F(x+h)-F(x)-BAh=Bo_1(h)+o_2(\phi(h))$ in detail

Please explain the steps involved after $F(x+h)-F(x)-BAh=Bo_1(h)+o_2\phi(h)$ in detail. Note 1:here differentiable means Frechet differentiable Source:Analysis for Applied Mathematics by WARD CHENEY(...
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### Reference Needed - Taylor's Theorem with Fréchet Derivatives

According to Wikipedia, Taylor's Theorem holds for Fréchet derivatives, but no reference is given. I started looking in various books and they all mention that it is possible to write down a version ...
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### Fréchet derivative of a matrix expression [closed]

Suppose $h(Q) = Q^{T} A Q$, then the Fréchet derivative is given by $D_{h} (Q) [H] = H^{T} A Q + Q^{T} A H$. I am bit unsure about this so-called Fréchet derivative is obtained. I would have just said:...
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### Meaning of delta-u term in variational derivative

I am considering a question in the calculus of variations. I can understand the concept of the variational derivative, but I am not sure what the $\delta u$ in this question means: "6. Let $Ω⊆R^n$...
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Suppose $X$ and $Y$ are Banach spaces and $U\subseteq X$, $V\subseteq Y$ are open subsets. Let $f:U\to V$ be bijective and continuously Fréchet differentiable with derivative $Df:U\to\mathcal{L}(X;Y)$....
### Is $n$-times differentiable equivalent to $n$-times Fréchet-differentiable for functions from $\mathbb C$ or $\mathbb R$ to a Banach space?
In the following, $\mathbb K$ denotes $\mathbb C$ or $\mathbb R$ and $E$ is a $\mathbb K$-Banach space I have known that for a function $f:\mathbb K\supseteq X\to E$ is called differentiable at $a$ ...