Given $F[u]$ and $G[v]$ are functionals of a real-valued function, what is $$ \int \frac{\delta F}{\delta u} \frac{\delta G}{\delta v} \, dx \quad ? $$

I have encountered such expressions for example in "Hamiltonian description of the ideal fluid" by P. J. Morrison. There Poisson brackets for several systems (mostly some version of ideal fluid) are given, like $$ \{F, G\} = \int \left[\frac{\delta F}{\delta q} \cdot \frac{\delta G}{\delta \pi} - \frac{\delta G}{\delta q} \cdot \frac{\delta F}{\delta \pi} \right] \, d a^3 \qquad (119) $$ $$ \{F, G\} = - \int_D \frac{\delta F}{\delta u} \frac{\partial}{\partial x} \frac{\delta G}{\delta u} \, d x \qquad (156) $$ and, in general: $$ \{F, G\} = \int_D \frac{\delta F}{\delta \psi^i} \mathfrak{J}^{ij} \frac{\delta G}{\delta \psi^j} \, d \mu \qquad (169) $$ where $F$ and $G$ are functionals of appropriate functions, $\mathfrak{J}$ is a certain operator.

Unfortunately it is not said what does it mean, or at least I overlooked it. Moreover, the biggest problem is that even types of expressions involved are not specified. I've tried to filled that gap.

Considering functions $u,v : M$ and functionals $F, G : M \to R$, $\; M$ being a manifold of certain kind of functions, functional derivative $D$:

$$D : (M \to R) \to M \to M \to R $$ $$ \left( \left(D \, F \right)\, u \right) \, [v] = \left[\frac{d}{d \varepsilon} F[u + \varepsilon v] \right]_{\varepsilon = 0}$$

thus $\delta F / \delta u$ stands likely for

$$(D \, F)(u) : M \to R.$$

Now $$ \int_D \frac{\delta F}{\delta u} v \, d x := \left< \frac{\delta F}{\delta u}, v \right> := \left[\frac{d}{d \varepsilon} F[u + \varepsilon v]\right]_{\varepsilon = 0}, $$ that is to some sort of contraction, analogous to covector and vector, and integral notation is just a reference to a finite-dimensional case. Thus I believe

$$ \int_D \frac{\delta F}{\delta u} \frac{\delta G}{\delta v} \, dx := \left[ \frac{\delta F}{\delta u}, \frac{\delta G}{\delta v} \right], $$ where $[\,,]$ is some sort of scalar product. Indeed, I guess scalar product between functions is implied: $$ \left[u. v \right] := \int_D uv \, dx, $$ but I don't see from $$\langle \,, \rangle : M \to (M \to R) \to R $$ $$[\,,] : M \to M \to R $$ how $[\,,]$ can be lifted to act for two functional derivatives. That is I know the mapping from functions to functionals, but not vice versa. Probably given enough limitations on $M$ there is one? Maybe someone will point me an accessible, but rigorous introduction to a subject.



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