TITLE: Difference between functional and function.
QUESTION [23 upvotes]: 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?

REPLY [6 votes]: 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.
$\mu = \mathbb{E}_{\mathbf{F}}[X]$)