Examples of functions that are not functionals
Matthew Harrington 
Functionals are known as functions that give a number as an output. Are there examples of functions that do not give a number as an output? I find it strange that it is necessary to define the term functional. Functions have an argument or an input. They also have rules (e.g. step functions). And they give an output. If we were to provide the value of the input, we would get a number. The wikipedia article in functionals mentions that "the mapping of a function to the value of the function at a point is a functional." Are there examples of functions that cannot be functionals?
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$\begingroup$A functional, $f$, is a special type of function, from a vector space $V$ to its underlying field, $F$, we write this $f\colon V\to F$. (Note $F$ may not be numbers. A functional does not have to give out a number).
A function is a map between any two arbitrary sets. So an example of a function, that's not a functional is $f\colon \{23,30\}\to\{\text{Green},\text{ Curry}\}$ where $f(23)=\text{Green}$ and $f(30)=\text{Curry}$.
Functions are quite abstract and general.
$\endgroup$ $\begingroup$Yes. Let's think about a "function" that takes some polynomial $f(x)$ of degree $d$ and gives us back another polynomial $f'(x)$ of degree $d-1$. This function I'll write as $D$, and is the derivative (which you many know if you've learned calculus).
Formally, the derivative is a linear mapping (another word for "function") $D:P_d(\mathbb R)\to P_{d-1}(\mathbb R)$. Given some function $ax^3+bx^2+cx$, $D(ax^3+bx^2+cx) = 3ax^2+2bx+c$. This isn't a number, but is a polynomial.
Generically, we can make many other "functions that aren't functionals". Think about the two-variable function $f(x,y) = 7xy+y^2$. If we look at this when $x = 2$, this gives us the function $g(y) = f(2,y) = 14y+y^2$. This act of substituting in $x = 2$ is the same as applying a function $H:P_d(\mathbb R)^2\to P_d(\mathbb R)$ to our initial function $f(x,y)$.
A mapping $F:A\to B$ will be a functional only if $B$ only has numbers in it (formally, is a field). As I've mentioned above, there are plenty of interesting mappings where $B$ isn't a field.
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