Frechét Derivative of F: $g \mapsto \int_0^n g(x)p(x)dx$

Question

I've been wondering what the Frechét Derivative of F: $g \mapsto \int_0^n g(x)f(x)dx$ is.

By definition, the Frechét Derivative $D_{F,g}(g)$ is such that (omitting limits on integrals):

$lim_{||g||\rightarrow 0}$ $\frac{\int (g+h)f - \int hf - D_{F,g}(g)}{||g||} = 0$

This yields a result which I wasn't expecting: $D_{F,g} = \int_0^n g(x)f(x)dx$. I feel like I've made a mistake somewhere.

Answer 1

This is no surprise at all. Your function $F$ is linear, so you'd expect its first order Taylor expansion to be $F$ itself, that is, have no approximation error. The Frechét derivative is, in effect, the coefficient of the "approximating" linear function. Said another way, would you be surprised if $f:\mathbb R\to\mathbb R$ given by $x\mapsto 3x$ turns out to have derivative $3$?