I have two questions about the relation between Gâteaux and functional derivatives.
In calculus of variation textbooks, the first variation is usually defined as follows. Let $V$ be a 'function space' and $f: V \to \mathbb{R}$ a functional. Then, the first variation of $f$ is defined by: \begin{eqnarray} \delta f|_{y}(\eta) \equiv \frac{d}{dt}\bigg{|}_{t = 0}f(y+t\eta) := \lim_{t\to 0}\frac{f(y+t\eta)-f(y)}{t} \tag{1}\label{1} \end{eqnarray} The above definition (\ref{1}) is just the usual definition of a Gâteaux derivative of a function $f$. As far as I know, Gâteaux derivatives are defined, in its most generic way, on locally convex spaces.
Question 1: If the Gâteaux derivative is defined on locally convex spaces and if the first derivative is defined as the Gâteaux derivative of a functional $f$, why do most books not assume $V$ to be a locally convex space and, instead, use an imprecise "function space"? At least for most purposes, aren't locally convex spaces enough to cover all possible function spaces used in this subject?
For the second question, the physicist's notion of a functional derivative is very close to the notion of a first variation defined above. In the calculus of variations, we are usually aiming to study functionals which have the form of an integral, so what physicists call functional derivative is the kernel of the integral obtained after evaluating the first derivative (in the physics literature, this kernel is defined as taken the derivative in the direction of a Dirac delta distribution).
Question 2: Assume that the function $f$ has an integral form. Is it correct to define the functional derivative as the function (if it exists) $\delta f/\delta y$ satisfying: \begin{eqnarray} \frac{d}{dt}\bigg{|}_{t = 0}f(y+t\eta) = \int \frac{\delta f}{\delta y}(x)\eta(x) dx \tag{2}\label{2} \end{eqnarray} or is there a better definition?