# Questions tagged [gateaux-derivative]

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75 questions
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### Can a variational inequality (necessary condition for minima) be strict?

A well-known fact in optimization theory is the following: Let $C_{ad}$ be a non-empty, convex subspace of a real Banach space $B$ and let $F: U \mapsto \mathbb{R}$ be a function defined on an open ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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 ...