Power series for functionals and notation for functionals I am trying to learn some functional analysis/calculus of variations, mainly for being able to perform functional derivatives on simple functionals found in physics (therefore I will not be too concerned -at least for the moment- about many mathematical details: I will get more functional analysis in my studies but in a couple of years, now I am just trying to learn something on my own).
I am looking for clarification to:


*

*different use of notation that I have found so far

*"functional" Taylor expansions

*chain rule for functionals


and I wonder if I can get some help here.
1a.  A functional is a rule that associates a number with a set of functions, say $F[f_1,f_2, ...]$. 
Suppose I have (simple case):
$\displaystyle F[f] = \int f(x)\,dx$
it it correct to write $F[f(x)]$? And is this equivalent to write $F[f](x)$? (x scalar).

Edit
I think I need the limits of integration here. So it would be:
$\displaystyle F[f](a,b) = \int^{a}_{b} f(x)\,dx$
right?

1b. I guess in the same way as a functional can depend on several functions, it can also depend on several independent variables. For example, if $F$ depends on the function $f_1(x)$ and on the independent variable $y_1$, is this a good notation to use?
$F([f_1](x);y_1)$ 
The use of a clear notation seems important when one has to finally perform variations. 
2a. Let's consider again $F[f](x)$ (x scalar). If I am asked to "expand it in powers of $f$" (text of an exercise) I would use something similar to a Taylor expansion of ordinary calculus:
$\displaystyle \delta F = \sum_{n} \frac{1}{n!} \int dx_1 dx_2 ...dx_n \frac{\delta^{n} F[f]}{\delta f(x_1) \delta f(x_2) ...\delta f(x_n)} \delta f(x_1) \delta f(x_2)... \delta f(x_n) $ 
To first order (around for example $f(x_0)$)
$\displaystyle F[f] \approx f[f(x_0)] + \int dx_1 + \frac{\delta F[f]}{\delta f(x_1)}\Big{|}_{f(x_0)} \delta f(x_1)$ 
but what would $\delta f(x_1)$ be? Something like $f(x) - f(x_0)$?
2b. Say that I have another functional: 
$F[n](\textbf{x} - \textbf{a})$ 
where $\textbf{x}$ and $\textbf{a}$ are vectors (in 3D) and $n(\textbf{x-a},t)$.
What does it mean to "expand it in powers of $\textbf{a}$"?
Since $f$ is a function of $\textbf{a}$, am I required to apply some sort of chain rule to obtain the coefficients for the expansion? I have seen the following:
$\displaystyle \frac{\delta F[G[f]]}{\delta f(x)}  = \int dx \frac{\delta F[G[f]]}{\delta G[f]}\frac{\delta G[f]}{\delta f(x)}$.
So in my case I would end up with (to first order in $\textbf{a}$ and omitting the variable $t$ that does not seem to play any role):
$\displaystyle F[n](\textbf{x} - \textbf{a}) \approx n[\textbf{a}] + \int d \textbf{x} \frac{\delta F[n]}{\delta n(\textbf{x-a})}\frac{\partial_i n(\textbf{x} - \textbf{a}_i)}{\partial_i \textbf{a}_i} \textbf{a}$
So I end up with a functional derivative and a partial (directional) derivative to evaluate.
Is this correct or at least on the right track?
 A: Note that basicaly any transformation is formaly defined as some function. So you may define a function $F$ taking one function argument as $F(f) = ∫_a^b f(x) dx$ with some fixed numbers $a, b$. You can also let $a, b$ be free parametres, so you get a function of three parameters $F(f, a, b) := ∫_a^b f(x) dx$.
You can also adopt some special notation for emphasizing the type difference of variables, like typing $F[f]$ instead of $F(f)$ for function argument or $F_{ab}[f]$, $F_a^b[f]$, or $F(f; a, b)$ instead of $F(f, a, b)$. This is similar to writing $x_n$ instead of $x(n)$ for members of a sequence.
But remember that this is just syntax sugar for more convenient reading and that it should not collide with the standard notation. For example your suggestion $F[f](a, b)$ could be easily read as “apply functional $F$ on function $f$, getting a number and apply this number on arguments $a, b$”, which makes no sense.
Also $F[f(x)]$ would mean “apply function $f$ on scalar $x$ obtaining a number and apply functional $F$ on this number”, which make no sense. $F[f](x)$ means the right thing, i.e. applying a functional on a function and then evaluating the resulting function at a number.
