# 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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### Graphical intuition behind general Taylor formula

I am trying to understand the visual intuition behind this formula, beyond its formal demonstration. If I think for a moment that $X$ and $Y$ are both equal to $\mathbb{R}^2$, I visualize this ...
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### Is the inverse of an interior chart of a submanifold an immersion?

Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$, $\Omega$ be an open subset of $M$, $\phi$ be a $C^1$-diffeomorphism$^1$ from $\Omega$...
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### How do we norm the space of Fréchet differentiable functions from a Banach space $E$ to $\mathbb R$?

Quick question, I couldn't find an answer to: If $E$ is a $\mathbb R$-Banach space, and $C^1(E,\mathbb R)$ is the space of continuously Fréchet differentiable functions from $E$ to $\mathbb R$, how ...
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### Uniqueness of the Frechet Derivative: the role of $x \in int_X(T)$

I'm currently trying to learn some functional analysis as a way to improve my ability to read economic theory papers. I've come across what I thought was a simple proof but on reflection I don't think ...
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### Smooth function of infinitely many variables

Consider $\mathbb{R}^\mathbb{N}$, the set of real-valued sequences $(x_n)_{n \in \mathbb{N}}$, as a mere set. Let $f : \mathbb{R}^\mathbb{N} \to \mathbb{R}$ be a function, and suppose that $f$ is ...
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### The existence of the derivative of a Banach space valued function

In Evans' Partial Differential Equations $\S 7.1$, there is a motivation for definition of weak solution. \begin{align*} (1) \begin{cases} u_t + Lu = f \ &\text{in} \ U_T\\ u = 0 \ &\...
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### Functional derivative where the functional is not an integral

Let $\phi(f)=F(f(x_1),f'(x_2),...,f^{(n)}(x_{n+1}))$ be a functional defined on $C^n([a,b])$: here $F$ is a given (say) smooth function and $x_1,...,x_{n+1}$ are given points in $[a,b]$. What is the ...
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### Where is the $l_\infty$ norm Fréchet differentiable?

Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff ...
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### For what $p \in [1; +\infty)$ is $l_p$ a FD-space?

Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff ...
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### Is any Banach norm in $\mathbb{R}^n$ almost everywhere Fréchet differentiable?

Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff ...
Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff ...
Suppose $(V, \|\cdot\|_V)$ and $(W, \|\cdot\|_W)$ are two Banach spaces and $f: V \to W$ is some function. We call a bounded linear operator $A \in B(V, W)$ Fréchet derivative of $f$ in $x \in V$ iff ...