# Is there a branch of mathematics which studies functions of functions?

I'm not a mathematician, sorry...

My question is best explained with an example:

I have lots of functions (programs in a functional programming language) in which the domain and codomain are lists of integers. These functions, given lists of integers return other lists.

Then I have other kinds of functions in which the domain is the previous mentioned functions, and the codomain is a real number. These functions given one of the previous ones return an error measure (this error is some measure of the difference between the returned list and the given list, the one given as input, but in sorted order).

So my question is if there is a branch of mathematics that studies this kind of stuff, these "functions of functions".

Any help is much appreciated.

• I'm sure there are many others, but functional analysis comes to mind. – Git Gud Jan 9 '15 at 0:44
• Or operator algebras. You can consider spaces of functions, and the linear functions between these spaces. – voldemort Jan 9 '15 at 0:51
• For the more abstract minded, category theory with all of its morphisms, functors and natural transformations comes to mind. – Arthur Jan 9 '15 at 1:15
• Lambda calculus is another way of studying functions of functions. Every object is a lambda term, and each lambda term has an interpretation as a function that takes lambda terms as input and yields lambda terms as output. – MJD Jan 9 '15 at 1:53
• functions of functions are just functions, and all of mathematics studies functions. – Mister Benjamin Dover Jan 11 '15 at 11:19

In linear algebra and functional analysis, we can define linear functionals on the dual space $X^*$ of a vector space $X$, say with field $\mathbb{R}$, i.e. the space of bounded linear functionals $T:X \to \mathbb{R}$. For example, consider $X = \mathbb{R}^2$ and $x$ be a vector in $X$, say $x = (1,0)^T$. We can define a linear operator $\hat{x}: X^* \to \mathbb{R}$ by
$$\hat{x}(T) = T(x) \text{ for } T \in X^*$$
Note that $x$ is fixed while $T$ is the variable.
What makes this more interesting is the so called F. Riesz Representation Theorem, relating the space $X$ with the double dual space $X^{**}$. The easiest version is probably that of a Hilbert space $H$, stating that:
Every bounded linear functional $T \in H^*$ can be characterized by a unique element $y \in H$, in the sense that $$Tx = (x,y) \text{ for } y \in H$$ Moreover, it is an isometry $$\|T\|_{H^*} = \|y\|_H$$