A function on the reals maps any real to another [or maybe the same] real. Given some systematic way to order the inputs, you could describe the function as a v

A function on the reals maps any real to another [or maybe the same] real. Given some systematic way to order the inputs, you could describe the function as a vector lookup table with an infinite number of elements -- one output for each possible input. That vector describes a single point in an infinite-dimensional space. Thus every function from R to R is a single point in an infinite-dimensional space. Now you can use linear algebra to move these points around in the infinite-dimensional space, measure how far two points [functions] are from each other, etc. That's functional analysis. The linear operators that do this moving around and measuring are called functionals to indicate that they take functions as arguments. (Like higher-order functions in a programming language.) "Functional Analysis" is thus "The analysis of the objects known as functionals". Differentiation is an example of a functional.