If a functional takes a function as an argument, isn't that function also always a linear functional?

Question

For me this confusion arises from the fact that physics texts and mathematics for physicist textbooks tend to loosely describe functionals as "functions of functions," while if I look through Analysis and linear algebra textbooks I find that only linear functionals seemed to be defined - and in a way that seems more general - that is, a map to a number on a vector space

For example, Shelden and Axler have the following section on linear functionals: here

In Comparison I see the following definition in Courant and Hilbert's text: here Here it says the domain of a functional is an "space" of functions. Isn't this space a function space and aren't function spaces also always linear vector spaces?

Aren't these two definitions defining an equivalent thing?

I've also had trouble finding any mention or explanation of "non-linear functionals" where the closest thing I can find is this entry in the European encyclopedia of mathematics that seems to provide the most general definition. It would appear that such an object wouldn't be a "function of a function" either. So if I were to use to looser definition of "functions of functions" does this mean I'm necessarily also referring to linear functionals? https://encyclopediaofmath.org/wiki/Functional

Answer 1

A functional is defined to be a linear operator from a vector space to its underlying vector field. We usually do functional analysis on vector spaces where the vectors are themselves functions, however there's nothing saying it has to be. For instance, you can view $\mathbb{R}$ as a 1 dimensional real vector space, and a functional here would just be a map $f(x)=kx$.

Not all function spaces are necessarily vector spaces, they need to fit the vector space axioms. So it has to be closed under an addition operator, in which there's a 0 vector, addition is commutative and associative, there's an inverse, and that scalar multiplication plays nicely as well.

For instance, you can view matrices as functions, so take a look at the space of $n\times n$ invertible matrices, also known as the "General linear group". This isn't a vector space since its not closed under addition or scalar multiplication and there's no 0 vector.