Difference between functional and function.

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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 ?

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?

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3 Answers

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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.

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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.

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

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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.$\mu = \mathbb{E}_{\mathbf{F}}[X]$)

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