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?

• As far as I know, a functional is a mapping from a space of functions (on some variables) back to the space of one of the said variables. If you know what a dual space is then the space of functionals on functions from $\mathbb{R}$ is dual to said space of functions. For instance, the definite integral between two point on the real line of some function $f: \mathbb{R} \to \mathbb{R}$ is a mapping $f \to \mathbb{R}$. – user27182 Mar 9 '13 at 21:14
• Ok, so it looks - from the answer below - like functionals are a more general concept than I thought. As an example of an instance of a functional, though, you could imagine plotting two point in the plane, joined by some curve $f(x)$. Now if we wish to minimize the length of our curve then we would need to minimize some integral of $f(x)$. The thing we would be trying to minimise would be a mapping from the space of possible curves (the space of functions $f(x)$) to the space of their lengths (so probably to $\mathbb{R}$, which is where we probably would take $x$ from in the first place). – user27182 Mar 9 '13 at 21:19
• A picture of what I mean: storyofmathematics.com/images2/bernoulli_variations.gif – user27182 Mar 9 '13 at 21:20
• The question was difference between function and functional not definition of functional. I guess the difference is when we refer to functional we mean a set of possible functions; but when we say function we mean a specific function not a set of functions. – Creator Jan 30 '16 at 19:05
• @Creator Providing the definition of a "functional" makes very clear how different it is and isn't from a function... – Olivier Sep 7 '17 at 20:42

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.

• Is it given that a function is both the input and the output? (Happens to be true for both of the examples you give) – Joseph Garvin Mar 28 '19 at 1:47
• So, for the examples I gave, the output is a scalar, since I included a point p (for the derivative at p, a scalar) and a pair of points (a,b) (for the integral on [a,b], a scalar). I think most classical problems fall along similar lines, a function that eats functions and spits out scalars. But I'm sure the word "functional" has been applied to the situation you're thinking of, e.g. the functional that eats (differentiable) functions and spits out their derivatives (a function) :-) – Fried Brice Mar 28 '19 at 3:58

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.

• Hey, I've just added a reference and described the issue am having above in detail. Do look into the link to see if it adds more direction. Thanks. – qlinck Mar 9 '13 at 21:14
• @qlinck Thanks, I added a blurb. – rschwieb Mar 9 '13 at 21:41

The word functional carries some ambiguity to it due to its meaning which can change depending on the context/field of study. Wikipedia lists three cases where the meaning of word functional changes upon context.

• In linear algebra as already stated by previous answers the word functional refers to a linear mapping from a vectors space $$\mathbf{V}$$ into a field $$\mathbf{F}$$ of scalars.
• In analysis it means a mapping from a space $$\mathbf{X}$$ into the reals $$\mathbb{R}$$ or complex $$\mathbb{C}$$ numbers but the mapping doesn't have to necessarily be linear
• In computer science, it indicates higher order functions, simply put functions that take other functions as arguments
• In statistics, it might be a function $$f$$ mapping to a distribution $$\mathbf{F}$$ where $$f$$ acts as the parameters of the distribution function $$\mathbf{F}$$ also known as statistical functionals. (e.g. $$\mu = \mathbb{E}_{\mathbf{F}}[X]$$)
• This is very helpful – AimForClarity Dec 21 '20 at 14:57