Difference between functional and function.

I have come across the term 'functional'. How is a 'functional' different from a 'function'? The exact term I came across was 'statistical functional.'

In terms of the background, can you please focus on the first few lines followed by the four examples in http://sites.stat.psu.edu/~dhunter/asymp/fall2002/lectures/ln14.pdf ?

The definition is seeming to be quite generalized given the examples. Is there a crux of the definition that is not too general nor too restricted?

Solution 1:

The modern technical definition of a functional is a function from a vector space into the scalar field. For example, finding the length of a vector is a (non-linear) functional, or taking a vector and returning the 3rd coordinate (relative to some basis) is a (linear) functional.

But in a classical sense, functional is an antiquated term for a function that takes a function as input. For example, the function derivativeAt(p)(_) that takes a function f and returns f'(p) is a functional in the classical sense, as well as the function integralOver(a,b)(_) that takes a function f and returns the integral of f on [a, b].

Today, we'd call these higher-order functions in a Comp Sci setting, but in a Math setting we typically take to just calling them functions, or colloquially functionals in order to distinguish them from the other functions we are working with at the moment. I suspect that your statistics text might be using this classical version of the term colloquially.

Solution 2:

Typically, a functional is just a function of a set into a field $F$. The ones I know best are "linear functionals," where a functional is a linear function of an $F$ vector space into $F$.

I'm afraid I don't know about the "statistical" reference, I just know the above is true in functional analysis and some applied mathematics that I've read.

Added: Ah, I also forgot that sometimes people call "functions of functions" functionals.