Comment by ttoinou

I agree but we're using functions for different things here. Yes some specific families of functions can be treated as vector spaces. In this article it seems like the author is pretending to take all real->real functions and treating them as if they are a vector space, whatever the content of the functions, quote :

  we’ve built a vector space of functions

and later he admits it is impossible

  Ideally, we could express an arbitrary function f as a linear combination of these basis functions. However, there are uncountably many of them—and we can’t simply write down a sum over the reals. Still, considering their linear combination is illustrative:

They are uncountable because they are aleph1