“AND” is 0001 for 00 01 10 11. 2^4=16 binary Boolean functions, in ternary it blows up, but it’s not infinite.
Take a vector for normal space, [x, y, z]. We say each component of this vector is one dimension, so this one is 3D, and each of its three components can vary. Two such vectors are different if one or more components differ between them. Treating a function as a vector means treating each distinct possible input to the function as a distinct component.
For example, consider the integer function f(x) = x^2. This can be represented as the vector [..., 16, 9, 4, 0, 4, 9, 16, ...] Where the complete vector has as many components as integers. Since there's infinitely many integers, there's infinite components, so instead of 3D like the three component vector above, this vector is ∞D.
Any single function is representable in this way, so each distinct function has its own unique infinitely long vector.
So each different function is a different "point" in an infinite dimensional vector space.
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.