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If the function $f(u)(x)$ is given as $f(u) = \int_{\Gamma} g(x,y) u(y) dy$, what is the derivative $df/du$?

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Formally, your function $f$ is linear in $u$. Do you know what is the derivative of a (continuous) linear function? –  abatkai Jun 28 '13 at 19:50
    
Perhaps math.SE would be a better site for the question. –  Igor Khavkine Jun 28 '13 at 22:35
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migrated from mathoverflow.net Jun 29 '13 at 2:01

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1 Answer

The Gâteau derivative of $f$ at $u$ along $v$ is $$ Df_u(v) = f(v). $$ Note that the derivative does not depend on $u$. Hence it is also the Fréchet derivative of $f$. Note that we are implicitly assuming that the kernel $g$ is so that $f$ is a bounded linear map between the relevant function spaces you are interested in.

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