# 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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### 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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