Is the functional derivative a function or a functional

[Sonderval]

I am confused whether the functional derivative ($\delta F[f]/\delta f$) is itself a functional or whether it is only a function

The Wikipedia article is not very rigorous
https://en.wikipedia.org/wiki/Functional_derivative
but from the examples (like Thomas-Fermi density), it seems as if the derivative of a functional is a function, for example
$$\frac{\delta F[f]}{\delta f}= \int f^n(x) dx =n f^{n-1}(x)$$

However, I would expect it to be a functional in itself (in the same way that the derivative of a function is a function)
$$\frac{\delta F[f]}{\delta f}= \int f^n(x) dx =\int n f^{n-1}(x) dx$$

 

[fresh_42]

A functional as I understand it is any function from a vector space into a field. Hence, every functional is always a function, but not vice versa.

What is a functional in your opinion?
What do you mean by functional derivative; $y\longmapsto \dfrac{\partial F[y]}{\partial y(x)}$ or $F\longmapsto \dfrac{\partial F[y]}{\partial y(x)}$ or simply the result $\left. \dfrac{d}{d\varepsilon }\right|_{\varepsilon =0}F(y+\varepsilon \Phi)\;$ in which case I'd ask what the variable is?

 

[Sonderval]

I understand a functional to be a map from a space of functions to a number, as in my example above:
$$ F[f] = \int_a^b f^n(x) dx$$
A Functional gets a function as input and gives a number.

The functional derivative should (if I understand things correctly, which I probably don't) produce a new object from a functional (so the second of your options above) in the same way the derivative of a function produces a new object (the derivative function in 1D or the gradient function in a vector space).

My question is exactly that: What kind of object is the derivative of a functional, i.e., if I apply the "operator"
$$\partial/\partial f$$ to a functional $$F[f]$$, what is the result? A functional? An object that maps a function to a function (like a gradient maps a vector to a vector)?

 

[pasmith]

It depends. From the functional $I$ you create a new object, $\delta I$. Evaluating this at $y$ gives you the linear functional
$$\delta I[y] : h \mapsto \left.\frac{d}{d\epsilon} I[y + \epsilon h]\right|_{\epsilon = 0}.$$ Therefore $\delta$ is a linear operator which maps a functional to a function from the space of functions to the space of linear functionals.

 

[Sonderval]

Thanks. But is this $\delta I[h]$ the same as $\delta I[h]/\delta h$?
In analogy to functions, I would expect the first to be the equivalent of a total differential (in functional logic a variation) and the second to be equivalent to a derivative. I find the nomenclature quite confusing, to be honest.