What makes the author say that functions are infinite dimensional? Seems like the space of functions might be infinite dimensional but one function is usually not.
“AND” is 0001 for 00 01 10 11. 2^4=16 binary Boolean functions, in ternary it blows up, but it’s not infinite.
I think I understand it, let's see if I can explain it. Hopefully I'll say something useful.
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.
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".
The author lives in a context of real calculus, as such he declares that the field from the second section onwards will be reals. The functions over the field of booleans can be equally interesting including the Fourier transformation (multiplication in n log n, iirc)! But they are less intuitive and less known.
“AND” is 0001 for 00 01 10 11. 2^4=16 binary Boolean functions, in ternary it blows up, but it’s not infinite.