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I have been studying calculus of varitions and one of the first things I came across was the functional derivative. When I looked for additional information and resources on the Wikipedia page was that when the space of functions is a Banach space, the derivative is actually something called the Frechet derivative, which directly generalizes ordinary calculus.
A question then popped in my mind - can we define a space of functionals to be a Banach space? Of course, every function is a functional, so we could just put only functions therein, but I want to consider some nontrivial cases, preferably those, wherein the action functional $S[q]=\int^b_aL[t,q(t),\dot q(t)]$ is contained. And what will be the Frechet derivative of such space (if it exists)?

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  • $\begingroup$ There is the directional derivative $\nabla_u S[q] = \lim_{\epsilon \to 0} \frac{S[q+\epsilon u]-S[q]}{\epsilon}$, as a function of $u$ it is a linear map $X \to \mathbf{R}$, the second derivative is a linear map $X \times X \to \mathbf{R}$ and so on. $\endgroup$ – reuns Nov 4 '17 at 18:56